Nonrelativistic near-BPS corners of N=4 super-Yang-Mills with SU(1,1) symmetry
NNonrelativistic near-BPS corners of N = 4 super-Yang-Mills with SU (1 , symmetry Stefano Baiguera, Troels Harmark, Nico Wintergerst
The Niels Bohr Institute, University of CopenhagenBlegdamsvej 17, DK-2100 Copenhagen Ø, Denmark
Abstract
We consider limits of N = 4 super Yang-Mills (SYM) theory that approach BPS boundsand for which an SU(1 ,
1) structure is preserved. The resulting near-BPS theories becomenon-relativistic, with a U(1) symmetry emerging in the limit that implies the conservationof particle number. They are obtained by reducing N = 4 SYM on a three-sphere andsubsequently integrating out fields that become non-dynamical as the bounds are approached.Upon quantization, and taking into account normal-ordering, they are consistent with takingthe appropriate limits of the dilatation operator directly, thereby corresponding to Spin Matrixtheories, found previously in the literature. In the particular case of the SU(1 , |
1) near-BPS/Spin Matrix theory, we find a superfield formulation that applies to the full interactingtheory. Moreover, for all the theories we find tantalizingly simple semi-local formulations astheories living on a circle. Finally, we find positive-definite expressions for the interactionsin the classical limit for all the theories, which can be used to explore their strong couplinglimits. This paper will have a companion paper in which we explore BPS bounds for which aSU(2 ,
1) structure is preserved. a r X i v : . [ h e p - t h ] N ov ontents S reduction and near-BPS limits 4 N = 4 SYM on S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Bosonic SU(1 ,
1) limit - The simplest case . . . . . . . . . . . . . . . . . . . . . 82.3 SU(1 , |
1) limit - A first glance at SUSY . . . . . . . . . . . . . . . . . . . . . . 122.4 Fermionic SU(1 ,
1) limit - A subcase of SU(1 , |
1) . . . . . . . . . . . . . . . . . 182.5 PSU(1 , |
2) limit - The maximal case . . . . . . . . . . . . . . . . . . . . . . . . 19 , |
1) near-BPS theory . . . . . . . . . . . . . . . . . . . . 243.2 Quantization vs near-BPS limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 ,
1) SMT . . . . . . . . . . . . . . . . . . . . 345.3 Local formulation of SU(1 , |
1) SMT . . . . . . . . . . . . . . . . . . . . . . . . 355.4 Local formulation of PSU(1 , |
2) SMT . . . . . . . . . . . . . . . . . . . . . . . 38 S
40B Hamiltonian and conserved charges of N = 4 on S
42C Properties of spherical harmonics and Clebsch-Gordan coefficients 49D Algebra and oscillator representation 56References 60 N = 4 super-Yang-Mills (SYM) theory is conjectured to describe strings and gravity onAdS × S in its strongly coupled limit. Accessing this regime is a challenging task andnecessitates looking for limits of N = 4 SYM in which its dynamics simplify. For instance,in its planar limit one achieves a powerful integrability symmetry that enables to solve forthe spectrum in the strong coupling limit [1]. This can for instance be used to obtain theHagedorn temperature at any ’t Hooft coupling [2, 3]. However, the planar limit correspondsto non-interacting strings and gravitons. Thus, even by including corrections to the planarlimit, one would not be able to study phenomena like black holes that involve strong gravity.Recently a different approach was advocated [4]. The proposal is to consider certain non-relativistic corners that arise as near-BPS limits of N = 4 SYM [5]. In such limits, one can1aintain a finite number of colors of N = 4 SYM, and hence strong gravity, but instead thestringy and gravitational dynamics become non-relativistic [6–9].One can motivate the interest in the resulting non-relativistic theories from two pointsof view. One is that they reveal new insights into the dynamics of N = 4 SYM and henceinto the AdS/CFT correspondence. Another is that these new theories might provide newnon-relativistic realizations of the holographic principle that are important to study on theirown.In this paper, we continue the investigations of the non-relativistic corners of N = 4 SYMset out in [4]. Starting with N = 4 SYM on a three-sphere, we consider limits that zoom inclose to BPS bounds of the type E ≥ S + X i =1 ω i Q i , (1.1)where E is the energy, S one of the angular momenta and Q i , i = 1 , ,
3, are the threeR-charges of N = 4 SYM on a three-sphere. Moreover, ω i , i = 1 , ,
3, are three constants thatcharacterize the BPS bounds. One can equally well translate these inequalities to bounds onthe scaling dimensions for N = 4 SYM on flat space via the state-operator correspondence.The near-BPS limits we consider send the ’t Hooft coupling λ to zero while keeping [5]1 λ ( E − S − X i =1 ω i Q i ) fixed . (1.2)Starting with the classical action for N = 4 SYM on a three-sphere, we show using spherereduction that most of the massive modes on the three-sphere decouple, leaving a subset ofdynamical modes that survive the limit. However, some of the non-dynamical modes, whichwe show includes spherical modes of the gauge field of N = 4 SYM, can still contribute to theeffective interaction of the surviving dynamical modes. Using this procedure, with reductionon S and integrating out non-dynamical modes, one obtains a classical description of thesurviving modes, which is the classical description of the near-BPS theory corresponding tothe given BPS bound (1.1).These classical near-BPS theories provide the effective description of N = 4 SYM nearthe BPS bounds (1.1). We find that all such theories are non-relativistic, in that antipar-ticles decouple in the limit. Accordingly, one observes the emergence of a U(1) symmetrycorresponding to a conserved number operator.Upon quantization of the near-BPS theories, they result in quantum mechanical theories.As part of the quantization one finds self-energy corrections that are easily computable froma normal-ordering prescription. We show that the quantized near-BPS theories correspondto the Spin Matrix theories [5] that were found previously by considering the same near-BPSlimits (1.2) taken on the quantized theory of N = 4 SYM on a three-sphere, as described bythe dilatation operator [10–12]. Indeed, one obtains in this way theories with only a subsetof the states of N = 4 SYM on a three-sphere, as the rest have decoupled. Moreover, theinteraction is directly related to the one-loop dilatation operator of N = 4 SYM. This showsthat one can consistently quantize the near-BPS theories that we obtain in this paper.Due to the particular form of the BPS bound (1.1), the near-BPS/Spin Matrix theories thatwe consider here have SU(1 ,
1) symmetry, possibly as subgroup of a larger global symmetry. In2he free limit the spectrum gives a free energy that goes like temperature squared, indicatingthat the theories are effectively (1 + 1)-dimensional. Therefore, one would expect to findformulations as non-relativistic (1 + 1)-dimensional quantum field theories. Indeed, suchformulations exist, albeit not as fully local quantum field theories and with non-standardfeatures similar to positive energy ghost fields.In detail we consider four different BPS bounds (1.1) depending on the choice of ( ω , ω , ω ).In the case ( ω , ω , ω ) = (1 , ,
0) one obtains a scalar theory with SU(1 , × U(1) global sym-metry that resemble the positive momentum modes of a scalar field on a circle. Interestingly,the interactions in this case can be viewed as arising from the coupling to a non-dynamicalscalar field, resembling a gauge field. With ( ω , ω , ω ) = ( , , ) one finds instead a theorywith fermionic modes with the same global symmetry that can be formulated in terms of thepositive momentum modes of a chiral fermion on a circle.For ( ω , ω , ω ) = (1 , , ) one obtains a non-relativistic theory with SU(1 , | × U(1)symmetry that can be regarded as a combination of the two latter theories, with a bosonicand a fermionic field on a circle. This theory is supersymmetric and one can find a superfieldformulation in which the interactions arise from integrating out the super-multiplet of a non-dynamical gauge field. This is the case that we are considering in most detail in this paper,since it is simple to describe but at the same time it contains the bosonic and fermionic caseswith SU(1 , × U(1) global symmetry as subsectors. Finally, we also consider the maximalcase with ( ω , ω , ω ) = (1 , ,
1) in which one has a theory with two scalars and two chiralfermions on a circle with PSU(1 , | × U(1) global symmetry .These four near-BPS/Spin Matrix theories are interesting in their own right since theyare consistent limits of N = 4 SYM on a three-sphere that describes the behavior of N = 4SYM near a BPS bound, or, equivalently, near a zero-temperature critical point if one takesthe planar limit [5]. Indeed, it is intriguing that one obtains non-relativistic behavior in suchlimits.Another important reason to study them is that they have holographic duals. One seesthis as consequence of the AdS/CFT correspondence, since one can take the same near-BPSlimit on the string theory side of the correspondence [5–9, 13, 14]. The philosophy here isthat one can hope to solve this corner of the AdS/CFT correspondence, and then exploitthis to learn about the full correspondence. This goal was realized in case of the Hagedorntemperature [2, 3, 13] and it is also the spirit of the papers [15, 16].Alternatively, and even more interestingly, one can view the near-BPS/Spin Matrix the-ories as fully consistent and self-contained theories that realize the holographic principle.Indeed, this is supported by the fact that Spin Matrix theories in the planar limit reduces tonearest-neighbor spin chains that in a continuum limit are described by sigma-models. Re-cently, such sigma-models where interpreted as part of a class of non-relativistic sigma-modelwith a structure that resembles ordinary relativistic string theory, and with a new type ofnon-relativistic target space geometry called U(1)-Galilean geometry [7–9]. In this sense onecan claim to have shown the emergence of geometry from the Spin Matrix theories.The missing piece for having a full-fledged realization of the holographic principle is to seethe emergence of gravity. In this regard, interesting progress has been made on beta-functioncalculations [17–20] in the related non-relativistic SNC [21, 22] and TNC [7–9] string theories,3roviding the hope that a similar calculation is possible for the string-dual of Spin Matrixtheory that indeed possess a Galilean Conformal Algebra as local symmetry.This paper is organized as follows. In Section 2 we consider the four near-BPS limits ofclassical N = 4 SYM on a three-sphere. This uses the sphere reduction of N = 4 SYM of [23]explained in Appendix A and performed in detail in Appendix B. In Appendix C we exhibitrelevant Clebsch-Gordan coefficients and further properties of spherical harmonics.In Section 3 we quantize the near-BPS theory with SU(1 , |
1) symmetry and show explic-itly that the resulting quantum mechanical theory is the same as the SU(1 , |
1) Spin Matrixtheory limit of N = 4 SYM. This means that whether one first quantizes, and then takes thenear-BPS limit, yields the same quantum mechanical theory as if one does it in the oppositeorder. Also, it means that we found a highly efficient way to compute the one-loop dilatationoperator of N = 4 SYM.In Section 4 we find a momentum-space superfield formalism for the SU(1 , |
1) near-BPStheory, showing manifestly the supersymmetry of this theory. In addition it reveals a verysimple formulation of the interactions via a non-dynamical gauge-field multiplet.In Section 5 we discuss in detail how to find a local formulation of our four near-BPS/SpinMatrix theories. This reveals intriguing results that shows rather simple formulations of theinteractions, at least in the SU(1 , |
1) case and its two SU(1 ,
1) subsectors. At the same time,the theories are not fully local and exhibit a ghost-like behavior of the dynamical fields. Aspart of this we exhibit the algebraic structure of the global symmetry groups in Appendix D.Finally, we present our conclusions and outlook in Section 6. S reduction and near-BPS limits In this section we consider the classical Hamiltonian of N = 4 SYM on a three-sphere in thenear-BPS limits of the type (1.2) and show how to derive the classical effective Hamiltonianof the surviving degrees of freedom. We consider four different limits. In each limit onehas fields that decouple and do not contribute to the dynamics. However the gauge field,and in some cases also fermionic fields, can contribute to the resulting dynamics even if theydecouple as degrees of freedom and thus they need to be properly integrated out. This happensbecause the corresponding fields are sourced and can be understood in the same manner asthe nondynamical modes of the photon mediating the Coulomb interaction.We summarize with the scheme in Fig. 1 the main steps of the procedure that we willperform in Sections 2 and 3.After setting the stage of the computations in Section 2.1 we first review in detail inSection 2.2 the limit given only by bosonic modes with a global SU(1 ,
1) symmetry. Thiscase was considered previously in [4]. Then we proceed in Section 2.3 with the limit thatadds fermionic modes to this, providing a theory with SU(1 , |
1) symmetry which we showexplicitly in Section 4 to be supersymmetric. We proceed with a subcase of this with onlyfermionic degrees of freedom in Section 2.4. Finally, in Section 2.5 we consider the limitgiving the maximally possible amount of bosonic and fermionic modes, which has a globalPSU(1 , |
2) symmetry and hence has extended supersymmetry.In Section 3 we show that when one quantizes the four classical near-BPS theories that we4igure 1:
Pictorial description of the procedure performed in Sections 2 and 3 to find an effectivequantum Hamiltonian starting from N = 4 supersymmetric Yang-Mills on a three-sphere. obtain in Sections 2.2-2.5, one obtains bosonic SU(1 ,
1) Spin Matrix theory (SMT), SU(1 , | ,
1) SMT and PSU(1 , |
2) SMT. N = 4 SYM on S Our starting point is the classical action of N = 4 super-Yang-Mills theory compactified on athree-sphere S = Z R × S q − det g µν tr − F µν − | D µ Φ a | − | Φ a | − iψ † a ¯ σ µ D µ ψ A + g X A,B,a C aAB ψ A [Φ a , ψ B ]+ g X A,B,a ¯ C aAB ψ † A [Φ † a , ψ † B ] − g X a,b (cid:16) | [Φ a , Φ b ] | + | [Φ a , Φ † b ] | (cid:17) . (2.1)From this one can straightforwardly obtain the classical Hamiltonian H of N = 4 SYM on S by a Legendre transform. In the action (2.1) g is the Yang-Mills coupling constant, and weintroduced complex combinations of the real scalar fields transforming in the representationof the R-symmetry group SO(6) ’ SU(4) , defined as Φ a = √ ( φ a − + iφ a ) with a ∈ { , , } . The Weyl fermions ψ A with A ∈ { , , , } transform in the representation of SU(4) . Theaction is canonically normalized on the R × S background with the radius of the three-sphereset to unity. The field strength is defined as F µν = ∂ µ A ν − ∂ ν A µ + ig [ A µ , A ν ] , (2.2)and the covariant derivatives D µ as D µ Φ a = ∂ µ Φ a + ig [ A µ , Φ a ] , (2.3) D µ ψ A = ∇ µ ψ A + ig [ A µ , ψ A ] , (2.4)5here ∇ µ is the covariant derivative on the three-sphere, i.e. it contains the spin connectioncontribution for the fermions. The C aAB are Clebsch-Gordan coefficients coupling two rep-resentations and one representation of the R-symmetry group SU(4) . All the fields in theaction transform in the adjoint representation of the gauge group SU( N ).One can now decompose all the fields into spherical harmonics on S . For this, we followthe procedure and conventions of [23]. We have given the relevant details of this in AppendixA and B.Before we turn to the individual limits we first discuss the gauge field. In all four limits, thegauge field degrees of freedom will decouple on-shell. However, it contributes to the dynamicsexactly like an off-shell longitudinal photon does in QED and i ntegrating it out gives rise toan effective interaction of the surviving mode at order g . Since this is a feature that all foursectors share, we make a few remarks about it here.We will work in Coulomb gauge, corresponding to imposing ∇ i A i = 0 . (2.5)In our analysis below, it proves useful to first integrate out, i.e. solve for, all auxiliary degreesof freedom, here the temporal and longitudinal components of the gauge field. This procedureis standard, but since it is central to our arguments we display it here in some detail.To this end, we focus on the quadratic action for the gauge field, but we also include ageneric source to keep track of the correct constraint structure. We have S A = Z R × S q − det g µν tr (cid:18) − F µν − A µ j µ (cid:19) . (2.6)The canonical momenta areΠ = 1 p − det g µν δS A δ ˙ A = 0 , Π i = 1 p − det g µν δS A δ ˙ A i = F i , (2.7)yielding the Hamiltonian H A = Z R × S q − det g µν tr (cid:18)
12 Π i + 14 F ij − A ( ∇ i Π i + j ) + A i j i + η ∇ i A i (cid:19) , (2.8)where we have introduced a Lagrange multiplier η to enforce Coulomb gauge. We obtain theconstraints ∇ i Π i + j = 0 , ∇ i A i = 0 . (2.9)We have chosen to treat A as a Lagrange multiplier that enforces the Gauss’ law, and nolonger as one of the dynamical variables . Thus, we have two second class constraints, enoughto eliminate the remaining unphysical degrees of freedom.In order to solve the constraints (2.9), it proves useful to decompose all the fields intospherical harmonics on S (see Appendix A). Inserting all the decompositions into the Hamil- This is possible because this field has no dynamics (the canonical momentum is vanishing), and the non-trivial spatial dependence is encoded into the momentum Π i . H A = tr X J,m, ˜ m X ρ = − | Π Jm ˜ m ( ρ ) | + X ρ = ± ω A,J | A Jm ˜ m ( ρ ) | − χ Jm ˜ m (cid:18) i q J ( J + 1)Π Jm ˜ m (0) + j † Jm ˜ m (cid:19) + X ρ = − A Jm ˜ m ( ρ ) j † Jm ˜ m ( ρ ) − i q J ( J + 1) η † Jm ˜ m A Jm ˜ m (0) , (2.10)while the constraints (2.9) become2 i q J ( J + 1)Π Jm ˜ m (0) + j † Jm ˜ m = 0 , A Jm ˜ m (0) = 0 . (2.11)Since we can directly solve the constraints for A Jm ˜ m (0) and its symplectic partner Π Jm ˜ m (0) , wecan insert the solution into the Hamiltonian without changing the Poisson bracket. We thusobtain the unconstrained Hamiltonian H A = tr X J,m, ˜ m X ρ = ± (cid:18) | Π Jm ˜ m ( ρ ) | + 12 ω A,J | A Jm ˜ m ( ρ ) | + A Jm ˜ m ( ρ ) j † Jm ˜ m ( ρ ) (cid:19) + 18 J ( J + 1) | j Jm ˜ m | . (2.12)The form of the currents can now straightforwardly be reconstructed from the full N = 4Hamiltonian, and all further interactions can be restored. Instead of doing so in full generality,we will consider the near-BPS limit individually and reconstruct the interactions case by case,where they simplify considerably.We proceed now by considering the four near-BPS limits individually in the followingsubsections 2.2-2.5. In each case we will employ the following procedure1. Isolate the propagating modes in a given near-BPS limit from the quadratic classicalHamiltonian.2. Derive the form of the currents that couple to the gauge fields.3. Integrate out additional non-dynamical modes that give rise to effective interactions ina given near-BPS limit.4. Derive the interacting Hamiltonian by taking the limit.In all of the four near-BPS limits the single angular momentum S is turned on, correspondingto BPS bounds of the form H ≥ S + P i =1 ω i Q i where H is the Hamiltonian, S is one ofthe angular momenta and Q i , i = 1 , ,
3, are the three R-charges of N = 4 SYM on S .The coefficients ω i in front of the R-charges are given in the Table 1. A derivation of thesecoefficients can be found in [24]. For each case, the near-BPS limit is g → H − S − P i =1 ω i Q i g fixed . (2.13)Note that N is held fixed in this limit while g →
0. We find that the surviving degrees offreedom are described by a Hamiltonian H limit of the form H limit = L + ˜ g H int , H int = lim g → H − S − P i =1 ω i Q i g N , (2.14)where L is the Cartan charge of SU(1 , H int is the part of the Hamiltonian that describesthe interactions and ˜ g is the coupling constant of the resulting non-relativistic theory.7ectors SU(1 ,
1) bosonic SU(1 ,
1) fermionic SU(1 , |
1) PSU(1 , | P i =1 ω i Q i Q ( Q + Q + Q ) Q + ( Q + Q ) Q + Q Table 1: List of the combinations of the R-charges defining the limits of N = 4 SYM theorytowards BPS bounds H ≥ S + P i =1 ω i Q i . (1 , limit - The simplest case The first BPS bound we consider is H ≥ S + Q . As we shall see, the dynamical theorythat one obtains from the near-BPS limit (2.13) has a global SU (1 , × U (1) symmetry ofthe interactions. Free Hamiltonian and reduction of degrees of freedom
We start from the quadratic Hamiltonian H , in which interaction terms are omitted, andalso the R-charge Q and the angular momentum S of N = 4 SYM on S . These are allgiven in Appendix B. The propagating degrees of freedom can be extracted by considering thenear-BPS limit to lowest order in the coupling, which means we should set H − Q − S = 0.The left hand side reads H − S − Q = X J,m, ˜ m tr ( | Π Jm ˜ ma + i ( δ a + ˜ m − m )Φ † Jm ˜ ma | + ( ω J − ( δ a + ˜ m − m ) ) | Φ Jm ˜ ma | + X κ = ± X A =1 , (cid:18) ω ψJ + m − ˜ m − κ (cid:19) ψ † JM,κ,A ψ AJM,κ + X A =2 , (cid:18) ω ψJ + m − ˜ m + κ (cid:19) ψ † JM,κ,A ψ AJM,κ + X ρ = − , (cid:16) | Π Jm ˜ m ( ρ ) − i ( m − ˜ m ) A † Jm ˜ m ( ρ ) | + ( ω A,J − ( m − ˜ m ) ) | A Jm ˜ m ( ρ ) | (cid:17) ) , (2.15)with ω J = 2 J + 1, ω ψJ = 2 J + and ω A,J ≡ J + 2. Equating this expression to zero nowyields a set of conditions on the fields. First of all, since for the gauge field | m − ˜ m | ≤ J + 1,one finds A Jm ˜ m ( ρ ) = O ( g ) , Π Jm ˜ m ( ρ ) − i ( m − ˜ m ) A † Jm ˜ m ( ρ ) = O ( g ) . (2.16)Second, for the scalar field Φ we find for J = − m = ˜ m Π J, − J,J + iω J Φ † J, − J,J = O ( g ) , (2.17)and for all other eigenvalues of momentum ( m, ˜ m )Φ Jm ˜ m = O ( g ) , Π Jm ˜ m = O ( g ) . (2.18)The other two scalar fields satisfy for all possible values of ( m, ˜ m ) the conditionsΦ Jm ˜ m = Π Jm ˜ m = Φ Jm ˜ m = Π Jm ˜ m = O ( g ) . (2.19)For the fermions, non-trivial degrees of freedom would arise when we are able to make theprefactor of the quadratic terms in the fields to vanish. However, when κ = 1 ω ψJ = 2 J + 32 , | m | ≤ J + 12 , | ˜ m | ≤ J , (2.20)8here is no way to make the J –independent constant to vanish. The same phenomenonhappens with κ = − , with the roles of ( m, ˜ m ) exchanged. This tells us that in the bosonicSU(1 ,
1) sector we have for all choices of the indices (
A, κ ) the condition ψ AJM,κ = O ( g ) . (2.21)It is clear that each of the above constraints eliminates a dynamical degree of freedom fromthe theory, as one forfeits the choice of freely choosing initial conditions. Instead, the corre-sponding fields are entirely determined by the remaining degrees of the freedom, as encodedby the right hand sides of the above relations. We can make these explicit by demandingcompatibility with Hamiltonian evolution. It is simple to see that Eq. (2.18) weakly com-mutes, i.e. commutes on the constraint surface, with H , since no linear term in Φ and Π arepresent. The same holds for the first constraint in Eq. (2.16), for the scalars in (2.19) and forthe fermionic field in (2.21). On the other hand, the gauge field does appear linearly, namelythrough its coupling to the sources, as outlined in Eq. (2.12). Therefore, { H, Π Jm ˜ m ( ρ ) − i ( m − ˜ m ) A † Jm ˜ m ( ρ ) } ≈ ( ω A,J − ( m − ˜ m ) ) A † Jm ˜ m ( ρ ) + j † Jm ˜ m ( ρ ) . (2.22)Hence we impose the RHS side as a constraint to have a consistent Hamiltonian evolution.Finally, one can check that Eq. (2.17) does not generate additional requirements. We thusobtain the set of constraints A Jm ˜ m ( ρ ) = − ω A,J − ( m − ˜ m ) j Jm ˜ m ( ρ ) , Π Jm ˜ m ( ρ ) = 0 , (2.23)Φ Jm ˜ ma =2 , = 0 , Π Jm ˜ ma =2 , = 0 , (2.24)Φ Jm ˜ m = 0 , Π Jm ˜ m = 0 (except when J = − m = ˜ m ) , (2.25)Π J, − J,J + iω J Φ † J, − J,J = 0 . (2.26)Thus, the only dynamical degrees of freedom left are the modes Φ J, − J,J that obey the con-straint (2.26). Essentially, (2.26) is responsible for making the limiting theory non-relativisticas it decouples the anti-particles. Indeed, this condition relates the momentum with the com-plex conjugate of the field, implying that at the quantum level the field Φ n will annihilatea particle and the hermitian conjugate Φ † n will create it. As we explain below, this goes inhand with a U(1) global symmetry responsible for the conservation of particle number. Thisbehavior is standard in the non-relativistic low-momentum limit of QFTs [25]. Here we seethat the same phenomenon happens when focusing on a near-BPS limit of N = 4 SYM.Before we turn to the interactions, we consider the free part of the resulting Hamiltonian.The quadratic piece is simply obtained by inserting the constraint (2.26) into the quadraticHamiltonian (B.4). Before doing so, however, we note that (2.26) implies a change of brackets,since Φ J, − J,J and (Φ J, − J,J ) † no longer commute on the constraint surface. The Dirac bracketscan be straightforwardly worked out, yielding (with matrix indices suppressed) { Φ J, − J,J , (Φ J , − J ,J ) † } D = i ω J δ JJ (2.27)We make the redefinition Φ J = √ ω J Φ J, − J,J , (2.28)9n order to have a canonical normalization and to take into account that J both takes integerand half-integer values. The Dirac bracket (2.27) then becomes canonical n (Φ n ) ij , (Φ † n ) kl o D = iδ n,n δ il δ kj (2.29)With this, we obtain for the quadratic Hamiltonian H = S + Q = ∞ X n =0 ( n + 1) tr | Φ n | . (2.30)It is important to see how the SU(1 ,
1) symmetry emerges. Consider L = ∞ X n =0 (cid:18) n + 12 (cid:19) tr | Φ n | , L + = ( L − ) † = ∞ X n =0 ( n + 1) tr (cid:16) Φ † n +1 Φ n (cid:17) . (2.31)Using the bracket (2.29) one finds that these charges obey the SU(1 ,
1) brackets { L , L ± } D = ± iL ± , { L + , L − } D = − iL . The interactions that we find below have vanishing brackets with L and L ± which means that the interactions have a global SU(1 ,
1) symmetry.The difference between H and L is H = L + 12 ˆ N , ˆ N ≡ ∞ X n =0 tr | Φ n | (2.32)We notice that ˆ N commutes with H , L and L ± as well as the interaction terms with respectto the bracket (2.29). This means in particular that ˆ N is a conserved charge. Indeed, ˆ N is thenumber operator when quantizing this theory, and we recognize the fact that the conservationof the number operator is a hallmark of a non-relativistic theory. This in turn means we areallowed to switch the free part of the Hamiltonian to be L instead of H since they differ bya conserved quantity (one can view this switch as a time-dependent redefinition of the fields).This will turn out to be a natural choice for all of the four limits. Interactions
We now exhibit the interacting part of the Hamiltonian H int that arise in the near-BPS limit.Since H − S − Q = 0 by construction, we can define the interacting Hamiltonian as H int = lim g → H − S − Q g N . (2.33)Non-trivial contributions to H int arise from integrating out the gauge field. On the surfacedefined by the constraints (2.23) we find that the contributions to H − S − Q amount to X J,m, ˜ m tr J ( J + 1) | j Jm ˜ m | − X ρ = ± ω A,J − ( m − ˜ m ) ) | j Jm ˜ m ( ρ ) | . (2.34)To these terms we should add contributions from the scalar sector that we will derive now. Tothis end, we add the entire scalar sector and work out the form of the currents. The relevantinteraction terms involving scalars in the Hamiltonian H of N = 4 SYM, Eq. (B.40), are X J,m, ˜ m tr ( g C J J ,JM C J J ,JM [Φ J , Φ J † ][Φ J , Φ J † ] + ig C J J ,JM χ JM (cid:16) [Φ J † , Π J † ] + [Φ J , Π J ] (cid:17) − g q J ( J + 1) D J J ,JMρ A JM ( ρ ) [Φ J , Φ J † ] (cid:27) . (2.35)10here we used the short-hand notation J = ( J, − J, J ) , (2.36) i.e. Φ J = Φ J, − J,J , since we can restrict ourselves to the surviving scalar modes. In thisexpression the quantities C , D are Clebsch-Gordan coefficients that couple respectively threescalars or two scalars and one vector harmonics. We define them and show some of theirproperties in Appendix C. From (2.35), we can directly read off the currents. We have j † Jm ˜ m = ig C J J ,JM (cid:16) [Φ J † , Π J † ] + [Φ J , Π J ] (cid:17) = 2 g (1 + J + J ) C J J ,JM [Φ J , Φ J † ] , (2.37)where the latter equality holds on the constraint surface. Furthermore j † Jm ˜ m ( ρ ) = − g q J ( J + 1) D J J ,JMρ [Φ J , Φ J † ] . (2.38)We can now proceed to find the interaction Hamiltonian (2.33). Employing (2.33) we obtain H int = 14 N X J,m, ˜ m X J ,J ,J ,J Y i =1 √ ω J i ! tr J ( J + 1) (1 + J + J )(1 + J + J ) C J J ,JM C J J ,JM − X ρ = ± ω A,J − ( m − ˜ m ) q J ( J + 1) q J ( J + 1) D J J ,JMρ ¯ D J J ,JMρ + 12 C J J ,JM C J J ,JM ! [Φ J , Φ † J ][Φ J , Φ † J ] . (2.39)where we used the redefinition (2.28). It is clear that the only nontrivial contributions arisefrom ˜ m = − m . For notational convenience, we consider M = ( − m, m ). Inserting this andmaking ω A,J explicit yields H int = 18 N X J,m X J ,J ,J ,J Y i =1 √ ω J i ! tr (cid:18) (1 + J + J )(1 + J + J ) J ( J + 1) + 1 (cid:19) C J J ,JM C J J ,JM − X ρ = ± J + 1) − m q J ( J + 1) q J ( J + 1) D J J ,JMρ ¯ D J J ,JMρ ! [Φ J , Φ † J ][Φ J , Φ † J ] . (2.40)We can now directly use the crossing relations (C.20) to see that upon a shift J → J − ρ = 1, all terms in the above sum cancel except for a nontrivial remainderfrom the lower boundary of summation. We distinguish between ∆ J ≡ J − J = 0 and∆ J = 0 and moreover choose J > J without loss of generality, accounting for the conversewith a factor of 2. In this way we obtain H ( J = J )int = 14 N tr X J ,J ≥ X ∆ J> J [Φ J , Φ † J +2∆ J ][Φ J +2∆ J , Φ † J ] = 12 N ∞ X l =1 l tr (cid:16) q † l q l (cid:17) , (2.41)where we defined the SU( N ) charge density q l = ∞ X n =0 [Φ † n , Φ n + l ] . (2.42)11ext, let us consider the case J = J . Here, the abovementioned trick to shift part of theexpression by J → J − J = 0 , and then it is summed over J > . Collecting everything, we get H ( J = J )int = 18 N tr X J ,J ≥ " − J J (1 + 2 J )(1 + 2 J ) + X J> J − (2 J − J )(2 J − J )( J + 1)(2 J + J + 1)(2 J + J + 1) ! × (2 J )!(2 J )!(2 J − J )!(2 J + J )! (2 J )!(2 J )!(2 J − J )!(2 J + J )! [Φ J , Φ † J ][Φ J , Φ † J ] , (2.43)which resums into H ( J = J )int = 18 N ∞ X n =0 − n n tr (cid:16) q [Φ † n , Φ n ] (cid:17) , (2.44)where q = P ∞ n =0 [Φ † n , Φ n ] is the SU( N ) charge. The Gauss law on the three-sphere impliesthat q = 0 and hence H ( J = J )int is zero when taking this into account.The full Hamiltonian (2.14) then becomes H limit = L + ˜ g N ∞ X l =1 l tr (cid:16) q † l q l (cid:17) (2.45)taking into account that all physical configurations have zero SU( N ) charge q = 0 due tothe Gauss law on the three-sphere and with L given in Eq. (2.31). This is the interactingHamiltonian describing the effective dynamics of N = 4 SYM near the SU(1 ,
1) bosonic BPSbound. It is a non-relativistic theory because of Eq. (2.26), which relates the canonical mo-mentum to the complex conjugate field, as it happens for this class of quantum field theories.In addition, the non-relativistic nature of the system is also clear from the conservation of thenumber operator ˆ N defined in (2.32) corresponding to a further U(1) symmetry in additionto the global SU(1 , ,
1) charges L and L ± (2.31) under the brackets (2.29). This shows that the interactionterm of (2.45) is invariant under a global SU(1 ,
1) symmetry. Upon quantization, we show inSection 3 that the Hamiltonian (2.45) is equivalent to SU(1 ,
1) Spin Matrix theory. (1 , | limit - A first glance at SUSY We turn now to the BPS bound H ≥ S + Q + ( Q + Q ). In this case, the theory thatemerges from the limit (2.13) has a SU(1 , | × U(1) symmetry of its interactions. As weshall see, the additional symmetry compared to the SU(1 ,
1) case of Section 2.2 is related tothe fact that one has fermionic modes in addition to the bosonic modes of the SU(1 ,
1) case.In Sections 3, 4 and 5 we study this theory further, and show among other things that it issupersymmetric. 12 ree Hamiltonian and reduction of degrees of freedom
We follow the same procedure as in Section 2.2. Using Appendix B we find that the quadraticterms in the left-hand side of the BPS bound H − S − Q − ( Q + Q ) ≥ H − S − Q −
12 ( Q + Q ) = X JM tr ( X κ = ± " ( ω ψJ + m − ˜ m − κ ) ψ † JM,κ, ψ JM,κ + X A =2 , (cid:18) ω ψJ + m − ˜ m + κ (cid:19) ψ † JM,κ,A ψ AJM,κ + ( ω ψJ + m − ˜ m ) ψ † JM,κ, ψ JM,κ + (cid:12)(cid:12)(cid:12) Π JM + i (1 + ˜ m − m )Φ † JM (cid:12)(cid:12)(cid:12) + ( ω J − (1 + ˜ m − m ) ) | Φ JM | + X a =2 , (cid:12)(cid:12)(cid:12)(cid:12) Π JMa + i (cid:18)
12 + ˜ m − m (cid:19) (Φ † a ) JM (cid:12)(cid:12)(cid:12)(cid:12) + ω J − (cid:18)
12 + ˜ m − m (cid:19) ! | Φ JMa | ! + X ρ = − , (cid:16) | Π Jm ˜ m ( ρ ) − i ( m − ˜ m ) A † Jm ˜ m ( ρ ) | + ( ω A,J − ( m − ˜ m ) ) | A Jm ˜ m ( ρ ) | (cid:17) ) . (2.46)Imposing that this expression is zero gives a set of constraints. While the combination ofR-charges is different from the bosonic SU(1 ,
1) case of Section 2.2, the constraints for thescalars are given by the same set (2.17), (2.18) and (2.19) because of the inequalities | m | ≤ J and | ˜ m | ≤ J. On the other hand, we have now surviving fermionic modes, corresponding tothe conditions A = 1 , κ = 1 , m = − J − , ˜ m = J . (2.47)All the other fermionic modes, i.e. modes with different SU(4) index A , different value of κ ,or with a different choice of momenta m, ˜ m , decouple in the g → A Jm ˜ m ( ρ ) = − ω A,J − ( m − ˜ m ) j Jm ˜ m ( ρ ) , Π Jm ˜ m ( ρ ) = 0 , (2.48)Φ Jm ˜ ma =2 , = 0 , Π Jm ˜ ma =2 , = 0 , (2.49)Φ ( J,m = − J, ˜ m = J )1 = 0 , Π ( J,m = − J, ˜ m = J )1 = 0 , (2.50)Π J, − J,J + iω J Φ † J, − J,J = 0 , (2.51) ψ A =1( J,m = − J − , ˜ m = J ); κ =1 = 0 , ψ A =1 J,m, ˜ m,κ = − = 0 , ψ A =2 , , Jm ˜ m,κ = 0 . (2.52)We notice again that (2.51) induces a change of the Dirac brackets for the bosonic modes asin Eq. (2.27), while this does not happen for the fermionic modes. For this reason we againuse the redefinition of the scalar modes (2.28). According to this, we defineΦ J ≡ √ ω J Φ J, − J,J , ψ J ≡ ψ A =1 J, − J − ,J ; κ =1 , (2.53)13hich correspond to the surviving degrees of freedom in the g → n ( ψ n ) ij , ( ψ † n ) kl o D = iδ n,n δ il δ kj (2.54)Evaluating now the free Hamiltonian H of Eq. (2.46) on the constraints (2.48)-(2.52) we find H = tr ∞ X n =0 (cid:20) ( n + 1) | Φ n | + (cid:18) n + 32 (cid:19) | ψ n | (cid:21) , (2.55)We record also the SU(1 ,
1) charges L = tr ∞ X n =0 (cid:20)(cid:18) n + 12 (cid:19) | Φ n | + ( n + 1) | ψ n | (cid:21) , (2.56) L + = ( L − ) † = tr ∞ X n =0 (cid:20) ( n + 1) Φ † n +1 Φ n + q ( n + 1)( n + 2) ψ † n +1 ψ n (cid:21) . (2.57)We notice that H is related to L as H = L + 12 ˆ N , ˆ N ≡ tr ∞ X n =0 ( | Φ n | + | ψ n | ) (2.58)ˆ N commutes with H , L and L ± as well as the interaction terms in terms of the brackets (2.29)and (2.54) thus ˆ N is a conserved charge. As for the bosonic SU(1 ,
1) case this corresponds tothe conservation of particle number and it gives an extra U(1) symmetry which is a signatureof non-relativistic theories. Moreover, we can again switch the free part of the Hamiltonianto be L instead of H since they differ by a conserved quantity. Interactions
Following the general steps given in Section 2.1 we need to derive the currents which couple tothe matter fields and to integrate out non-dynamical modes which give non-vanishing quarticeffective interactions. The interacting Hamiltonian in this limit is defined by H int = lim g → H − S − Q − ( Q + Q ) g N . (2.59)The contribution to H − S − Q − ( Q + Q ) of the currents that couple to the gauge fieldis again given by Eq. (2.34). In addition to this, one has the interaction terms recorded inEq. (2.35), and from Eq. (B.40), one sees that one has interactions between the survivingfermionic modes and the modes of the gauge field as well X JM X J ,J tr (cid:16) g F ¯ J ¯ J ,JM χ JM { ψ † J , ψ J } + g G ¯ J ¯ J ,JMρ A JM ( ρ ) { ψ † J , ψ J } (cid:17) , (2.60)where we introduced the short-hand notation ¯ J = ( J, J + 12 , − J, κ = 1) (2.61) The F and G Clebsch-Gordan coefficients are evaluated on the momenta corresponding to the survivingdynamical degrees of freedom of the sector. However, due to the redefinition (B.8) of the fermionic modeswith κ = 1, which exchanges a field with its hermitian conjugate, the momenta on which the Clebsch-Gordancoefficients (2.61) are evaluated have opposite signs than the momenta of the modes (2.53). Thanks to thismodification, we observe that the conditions on momenta coming from triangle inequalities of the Clebsch-Gordan coefficients F , G are consistent with momentum conservation as evaluated directly from the creationor annihilation of particles dictated by the field content of the interactions. F and G . Finally, thereare Yukawa-type terms that couple fermions and the scalar fields − i g √ X J,J ,J (cid:26) ( − m − ˜ m + m − ˜ m + κ F J ,M ,κ J , − M ,κ ; J, − M (cid:16) ψ † J ,M ,κ , [Φ † JM , ψ J , − M ,κ , ] (cid:17) − ( − − m + ˜ m + κ F J , − M ,κ J ,M ,κ ; J,M (cid:16) ψ J ,M ,κ [Φ JM , ψ † J , − M ,κ , ] (cid:17) − ( J ↔ J ) (cid:27) , (2.62)where the antisymmetrization J ↔ J in the last line is referred to all the terms in theinteraction. Using the properties of F written in (C.24) and (C.25) one finds − i g √ X J ,J X J,M,κ − J F J JMκ ; J n − ψ † J,M,κ, [Φ J † , ψ J ] + ψ J,M,κ [Φ J , ψ † J ] o . (2.63)With this, we recorded all the interaction terms in the Hamiltonian of N = 4 SYM on S which are relevant for this particular limit .From the terms (2.60) which couple to the gauge field we extract the currents j † Jm ˜ m = 2 g (1 + J + J ) C J J ,JM [Φ J , Φ J † ] + g F ¯ J ¯ J ,JM { ψ J , ψ † J } , (2.64) j † Jm ˜ m ( ρ ) = − g p J ( J + 1) D J J ,JMρ [Φ J , Φ J † ] + g G ¯ J ¯ J ,JMρ { ψ J , ψ † J } . (2.65)We can now use this in Eq. (2.34) to find the explicit contributions from integrating out thegauge field. The purely bosonic part, which combines with the quartic scalar self-interactionin Eq. (2.35), gives as a result the Eq. (2.40), and after solving the sum over J one findsEqs. (2.41) and (2.43).The purely fermionic part of Eq. (2.34) combined with (2.64)-(2.65) gives instead12 N X JM X J ,J ,J ,J tr − X ρ = ± ω A,J − ( m − ˜ m ) G ¯ J ¯ J ,JMρ ¯ G ¯ J ¯ J ,JMρ + 14 J ( J + 1) F ¯ J ¯ J ,JM F ¯ J ¯ J ,JM (cid:19) { ψ † J , ψ J }{ ψ † J , ψ J } , (2.66)where ¯ G denotes the complex conjugate of G and we divided with g N as in (2.59). Toevaluate this we use the results of Appendix C where we expressed all the above terms withClebsch-Gordan coefficients using only the C Clebsch-Gordan coefficient. It is important inthis to keep properly track of the constraints on the momenta. All terms in the sum shouldhave m = − ˜ m = J − J = J − J ≡ ∆ J . (2.67)On the other hand, the triangle inequalities fix the range of summation of the momentum J and slightly differ for the various terms. The quadratic combinations in F impose | J − J | ≤ J ≤ min( J + J , J + J ) while the quadratic expressions in G impose the same conditionswhen ρ = 1 , and the different constraints | J − J | ≤ J ≤ min( J + J − , J + J −
1) when ρ = −
1. A remarkable simplification comes now from applying Eq. (C.47), which upon ashift J → J − G ¯ G with ρ = − , allows to cancel all the terms in the The full interacting Hamiltonian of N = 4 SYM action after reduction on the three-sphere is given inEq. (B.40). J, the only remaining term coming exactly from G ¯ G with ρ = − J . In conclusion, (2.66) gives the followingfour-fermion contributions to H int N ∞ X l =1 l tr(˜ q † l ˜ q l ) − N ∞ X n =0 tr (cid:16) { ψ † n , ψ n } ˜ q (cid:17) , (2.68)where we defined the SU( N ) charge density˜ q l = ∞ X n =0 √ n + 1 √ n + l + 1 { ψ † n , ψ n + l } . (2.69)Note that the first term in (2.68) arise from ∆ J = 0. Instead the second term arise from∆ J = 0 and one sees that it is zero on singlet states and therefore does not contribute. Notealso that an important difference from the case ∆ J = 0 is that the F F term in (2.66) has asingular prefactor when J = 0 , which means that the only remaining contribution from theentire interaction comes from the G ¯ G term with ρ = 1 evaluated at J = 0.Finally, the mixed bosonic-fermionic part of Eq. (2.34) combined with (2.64)-(2.65) givesthe following contribution to H int N X JM X J ,J ,J ,J X ρ = ± (cid:16)p J ( J + 1) D J J ,JMρ ¯ G ¯ J ¯ J ,JMρ + p J ( J + 1) ¯ D J J ,JMρ G ¯ J ¯ J ,JMρ (cid:17) ω A,J − ( m − ˜ m ) + J + J + 14 J ( J + 1) (cid:16) C J J ,JM F ¯ J ¯ J ,JM + C J J ,JM F ¯ J ¯ J ,JM (cid:17)(cid:27) √ ω J ω J tr (cid:16) [Φ J , Φ † J ] { ψ J , ψ † J } (cid:17) . (2.70)Notice that the two pieces mediated by the gauge field come in pairs, with the constraints m = − ˜ m = ∆ J ≡ J − J = J − J ∨ m = − ˜ m = − ∆ J ≡ J − J = J − J . (2.71)For this reason, it is possible to split the result in two parts. For both of them, the sum over J can be analytically computed with a similar trick as in the previous cases: we shift the termswith ρ = − DG to find that the sum vanishes due to (C.57), and then weconclude that there is only a contribution from the lower extremum of summation. Collectingthese results and adding the quartic bosonic terms Eqs. (2.41) and (2.43) and quartic fermionicterms (2.68) one finds the simple result12 N ∞ X l =1 l tr (cid:16) ˆ q † l ˆ q l (cid:17) − N tr (cid:16) ˆ q (cid:17) + 14 N ∞ X n =0 n + 1 tr (cid:16) [Φ † n , Φ n ]ˆ q (cid:17) , (2.72)where we defined the total SU( N ) current density ˆ q l = q l + ˜ q l . This expression is the quarticscalar self-interaction of Eq. (2.35) plus Eq. (2.34) with the current given by (2.64)-(2.65).We see that only the first term in (2.72) contributes to the interactions since the Gauss lawon the three-sphere means that ˆ q is zero.Finally, we should consider the Yukawa-type terms (2.63). The presence of these interac-tions imply that the field ψ is sourced, and should thus be integrated out. After doing that,we find the following further contribution to H int N X J,M,κ X J ,J ,J ,J − J − J F ¯ J JMκ ; J F ¯ J JMκ ; J √ ω J ω J ( κω ψJ − ( m − ˜ m )) tr (cid:16) [Φ J , ψ † J ][ ψ J , Φ † J ] (cid:17) . (2.73)16e consider the sum over J by splitting between the cases J < J , J ≥ J . It turns out thatthe argument of the sum vanishes once we shift J → J − in the term with κ = − . Thenthe result reduces to a boundary term when J ≥ J , while it vanishes when J < J , since inthe two cases the extremes of summation change. The result is N X J ,J , ∆ J ≥ tr (cid:16) [Φ J , ψ † J +2∆ J ][ ψ J +2∆ J , Φ † J ] (cid:17)p (2( J + ∆ J ) + 1)(2( J + ∆ J ) + 1) , (2.74)where we defined ∆ J ≡ J − J = J − J .In summary, the effective Hamiltonian in the g → H ≥ S + Q + ( Q + Q ) is H limit = L + ˜ g H int (2.75)with the interaction Hamiltonian (2.59) given by H int = 12 N ∞ X l =1 l tr (cid:16) ˆ q † l ˆ q l (cid:17) + 12 N ∞ X l =0 tr (cid:16) F † l F l (cid:17) . (2.76)where we defined F l = ∞ X m =0 [ ψ m + l , Φ † m ] √ m + l + 1 . (2.77)In Eq. (2.76) we took into account that all physical configurations have zero SU( N ) chargeˆ q = 0 due to the Gauss law on the three-sphere. Note that H int is manifestly positive.One can check that the interacting Hamiltonian H int commutes with the number operatorˆ N of Eq. (2.58) as well as the SU(1 ,
1) charges L and L ± in (2.56)-(2.57) with respect tothe Dirac brackets (2.29) and (2.54). This means the theory has a global SU(1 , × U(1)invariance. However, this can be enhanced to SU(1 , | × U(1) by considering the conservedsupercharges. We define Q = ∞ X n =0 r n + 12 tr (cid:16) ψ † n Φ n +1 + Φ † n ψ n (cid:17) . (2.78)One can now show L = {Q , Q † } D , { H int , Q} D = 0 , (2.79)using the Dirac brackets (2.29) and (2.54). This reveals that the near-BPS theory is super-symmetric.The non-relativistic nature of the near-BPS theory is apparent from the the conservationof the number operator ˆ N , which is related to the decoupling of anti-particles in the limit asone can see from the constraint Eq. (2.51). In addition, it is seen by the fact that the surviv-ing dynamical fermion appears with only a fixed choice for the chirality κ = 1 , thus giving adescription in terms of a single Grassmann-valued field. This phenomenon also happens whenconsidering the non-relativistic limit of the Dirac equation in 3 + 1 dimensions, since after We observe that the physical consequence of having different extremes of summation is that the interactiononly contains a particular assignment of momenta: in the quantized theory we annihilate a boson and createa fermion with higher momentum, and at the same time we annihilate a fermion to create a boson with lowermomentum. The reverse possibility is forbidden. c → ∞ one of the Weyl spinors composing the Dirac fermion becomes heavy anddecouples from the theory, leaving only a single Weyl spinor entering the Schroedinger-Pauliequation. Such a result can be found by requiring Galilean invariance from first principles andcan be generalized to other dimensions [26]. It also applies in the context of null reduction,a procedure that allows to find Bargmann-invariant theories starting from relativistic sys-tems in one higher dimension [27]. This mechanism works naturally also for non-relativisticsupersymmetric theories built from null reduction [28].In Section 3 we quantize this theory and find that it is equivalent to SU(1 , |
1) Spin Matrixtheory. In Section 4 we show that the natural presence of the supercharge Q is related to thefact that one can formulate it in terms of a momentum-space superfield formalism. Finally,in Section 5 we consider a local formulation of this near-BPS theory and comment on this. (1 , limit - A subcase of SU (1 , | For completeness we consider here briefly the BPS bound H ≥ S + ( Q + Q + Q ). Thenear-BPS limit gives in this case a subsector of the SU(1 , |
1) near-BPS limit in which onlythe fermionic modes survive. The global symmetry of this theory is SU(1 , × U(1).Considering the quadratic terms on the left-hand side of the BPS bound H − S − ( Q + Q + Q ) ≥ H − S −
23 ( Q + Q + Q ) == X J,m, ˜ m tr ( (cid:12)(cid:12)(cid:12)(cid:12) Π Jm ˜ ma + i (cid:18)
23 + ˜ m − m (cid:19) Φ † Jm ˜ ma (cid:12)(cid:12)(cid:12)(cid:12) + ω J − (cid:18)
23 + ˜ m − m (cid:19) ! | Φ Jm ˜ ma | + X κ = ± (cid:16) ω ψJ + m − ˜ m − κ (cid:17) ψ † JM,κ, ψ JM,κ + X A =2 , , (cid:18) ω ψJ + m − ˜ m + κ (cid:19) ψ † JM,κ,A ψ AJM,κ + X ρ = − , (cid:16) | Π Jm ˜ m ( ρ ) − i ( m − ˜ m ) A † Jm ˜ m ( ρ ) | + ( ω A,J − ( m − ˜ m ) ) | A Jm ˜ m ( ρ ) | (cid:17) ) . (2.80)As in Sections 2.2 and 2.3 this provides a set of constraints. Comparing to Section 2.3 we findthat the constraints are (2.48), (2.49) and (2.52) with the additional constraintsΦ Jm ˜ m = Π Jm ˜ m = 0 , (2.81)which means that all scalar fields decouple. The only surviving modes are thus ψ J ≡ ψ A =1 J, − J − ,J ; κ =1 with the Dirac anti-bracket given by (2.54). Since the gauge field enters inthe same way as in the SU(1 , |
1) case of Section 2.3 the terms that one obtains from inte-grating out the gauge field are the same. Thus, one can obtain the interacting Hamiltonian H int of this near-BPS limit simply by be setting the modes Φ n = 0 in the SU(1 , |
1) case. Wefind therefore H limit = L + ˜ g H int with H int = 12 N ∞ X l =1 l tr (cid:16) ˜ q † l ˜ q l (cid:17) . (2.82)where the SU( N ) charge density is defined by (2.69) and we took into account that all physicalconfigurations have zero SU( N ) charge ˜ q = 0 due to the Gauss law on the three-sphere. The18roperties of this theory are now inherited from the SU(1 , |
1) case. In particular, H int hasthe global symmetry with respect to SU(1 , N is conservedwhich again is in accordance with this being a non-relativistic theory. (1 , | limit - The maximal case The last BPS bound that we consider is H ≥ S + Q + Q . The theory emerging from thelimit (2.13) contains interactions with global invariance PSU(1 , | × U(1) . In particular, itis supersymmetric and includes a SU(2) residue of the original R-symmetry: this means thatwe will find again both bosonic and fermionic modes, but now both of them will transform asa doublet under this group. In Section 3 we quantize this theory, while in Section 5 we showthat it can be described in terms of local fields.
Free Hamiltonian and reduction of the degrees of freedom
Given the free Hamiltonian H and the Cartan charges derived in Appendix B, we considerthe near-BPS bound at lowest order in the coupling, i.e. we impose H − S − Q − Q = 0 . The left hand side reads H − S − Q − Q =tr X JM ( X κ = ± " X A =1 ( ω ψJ + m − ˜ m )( ψ AJM,κ ) † ψ AJM,κ − κ ( ψ A =1 JM,κ ) † ψ A =1 JM,κ + κ ( ψ A =2 JM,κ ) † ψ A =2 JM,κ + X a =1 , | Π JMa + i (1 + ˜ m − m )(Φ † a ) JM | + ( ω J − (1 + ˜ m − m ) ) | Φ JMa | + | Π JM + i ( ˜ m − m )Φ † JM | + ( ω J − ( ˜ m − m ) ) | Φ JM | + X ρ = − , (cid:16) | Π Jm ˜ m ( ρ ) − i ( m − ˜ m ) A † Jm ˜ m ( ρ ) | + ( ω A,J − ( m − ˜ m ) ) | A Jm ˜ m ( ρ ) | (cid:17) ) . (2.83)The vanishing of this expression gives a set of constraints. The common feature with the othercases is that the gauge field is non-dynamical, since it appears again with the same combinationas in Section 2.2, and then gives rise to the same constraints (2.23). On the other hand, nowthere is more space for scalars and fermions, indeed we find the generalization of Eq. (2.26)Π J, − J,Ja + iω J Φ † J, − J,Ja = 0 ( a = 1 , , (2.84)and there are no constraints on the fermionic modes with A = 1 , κ = 1 , m = − J − , ˜ m = J , (2.85) A = 2 , κ = − , m = − J , ˜ m = J + . (2.86)All the other scalars and fermionic modes decouple in the g → i.e. no additional constraints are generated, except for the non-trivial Dirac bracket involving thenew scalar surviving the limit, in complete analogy with Eq. (2.27) { Φ J, − J,J , (Φ J , − J ,J ) † } D = i ω J δ JJ . (2.87)19he entire set of constraints is given by (2.48)-(2.52), the only difference being that we needto apply all the previous identities involving the scalar Φ and the fermion ψ to the newdynamical modes Φ , ψ , too . Indeed, the dynamical bosons and fermions form a doubletunder the residual SU(2) R-symmetry. We remark this explicitly, and we canonically normalizethe Dirac brackets of the scalar fields, by introducing the notationΦ Ja ≡ (cid:16) √ ω J Φ J, − J,J , √ ω J Φ J, − J,J (cid:17) , (2.88) ψ a J ≡ (cid:16) ψ A =1 J, − J − ,J,κ =1 , ψ A =2 J, − J,J + 12 ,κ = − (cid:17) . (2.89)They will be the dynamical modes entering all the interactions of the sector, with bracketsgiven by Eq. (2.29) and (2.54) for all the fields in each doublet.The evaluation of the free Hamiltonian H in Eq. (2.83) on the constraints gives H = tr ∞ X n =0 (cid:20) ( n + 1) | Φ an | + (cid:18) n + 32 (cid:19) | ψ an | (cid:21) , (2.90)which is the natural generalization of the quadratic Hamiltonian of the SU(1 , |
1) sector. TheSU(1 ,
1) generators similarly generalize with a SU(2) structure and read L = tr ∞ X n =0 (cid:20)(cid:18) n + 12 (cid:19) | Φ an | + ( n + 1) | ψ an | (cid:21) , (2.91) L + = ( L − ) † = tr ∞ X n =0 (cid:20) ( n + 1)(Φ † a ) n +1 Φ an + q ( n + 1)( n + 2)( ψ † a ) n +1 ψ an (cid:21) . (2.92)This shows that the free Hamiltonian and L are related with a shift by a number operatorˆ N such that H = L + 12 ˆ N , ˆ N ≡ tr ∞ X n =0 (cid:16) | Φ an | + | ψ an | (cid:17) . (2.93)The number operator ˆ N is a conserved charge because it commutes with H , L , L ± and theinteractions, due to the brackets (2.29) and (2.54). Hence we can define the free part of theHamiltonian to be L ; the charge ˆ N and the corresponding invariance of the Hamiltoniancorrespond to the particle number symmetry typical of non-relativistic theories. Interactions
The interacting Hamiltonian in this sector is defined by H int = lim g → H − S − Q − Q g N . (2.94)Following the general strategy outlined in Section 2.1, we identify the following interactions:• Contribution of the currents for the coupling to the non-dynamical gauge field.• Quartic scalar self-interaction. Strictly speaking, the identities involving the two fermions are not the same, because the dynamical modesdiffer. However, here we mean that all the fermionic modes vanish except for the cases selected by (2.85) and(2.86).
20 Yukawa term, which gives rise to effective quartic interactions after integrating out oneof the non-dynamical fields.In principle these possibilities are the same allowed for the SU(1 , |
1) sector, the differencebeing that from a technical point of view there are more possibilities among the non-dynamicalfields to integrate out, and the interactions have an additional SU(2) structure.We start from the generalization of the currents (2.64) and (2.65), which now read j † Jm ˜ m = g J + J + 1 √ ω J ω J C J J ,JM [Φ a J , (Φ a ) † J ]+ g F ¯ J ¯ J ,JM { ψ J , ( ψ ) † J } + g F ¯ J ¯ J ,JM { ψ J , ( ψ ) † J } , (2.95) j † Jm ˜ m ( ρ ) = − g s J ( J + 1) ω J ω J D J J ,JMρ [Φ a J , (Φ a ) † J ]+ g G ¯ J ¯ J ,JMρ { ψ J , ( ψ ) † J } − g G ¯ J ¯ J ,JM, − ρ { ψ J , ( ψ ) † J } . (2.96)Here we used Eq. (C.29) and (C.39) to express the result only in terms of the short-handClebsch-Gordan coefficients introduced in Eq. (2.36) and (2.61), and we immediately rescaledthe scalar fields according to the definition (2.89).These currents are singlets under SU(2) and contributes to the interactions via Eq. (2.34).Using techniques analog to the method explained in Section 2.3 for the SU(1 , |
1) sectorby means of the identities given in Appendix C, we reduce all the sums over intermediatemomenta J to a boundary term.In order to perform this method for the purely scalar part, however, we also need toinclude the quartic bosonic self-interaction, which partially contribute to this result. Thecorresponding term in the general N = 4 SYM Hamiltonian is g (cid:18)
12 ([Φ , Φ † ] + [Φ , Φ † ] ) + | [Φ , Φ ] | + | [Φ , Φ † ] | (cid:19) , (2.97)which we can equivalently write as g (cid:18) | [Φ a , (Φ † a )] | + | [Φ a , Φ b ] | (cid:19) (2.98)The first term contributes to the effective interactions mediated by the gluons, having thestructure of a product of SU(2) singlets, i.e. it has a SU(2) double trace structure. Combiningsuch a term with the formula (2.34) with currents (2.95) and (2.96), we obtain12 N ∞ X l =1 l tr (cid:16) ˆ q † l ˆ q l (cid:17) , (2.99)where the charge densities are ˆ q l = q l + ˜ q l with q s ≡ ∞ X n =0 X a =1 , [(Φ † a ) n , (Φ a ) n + l ] , ˜ q l ≡ ∞ X n =0 X a =1 , √ n + 1 √ n + l + 1 { ( ψ † a ) n , ( ψ a ) n + l } , (2.100)The other term included in the quartic scalar self-interaction (2.98) requires some additionalcare. It is a SU(2) single trace operator, and as such it cannot can be mediated by a SU(2)21inglet. Consequently, it gives rise to a genuinely new interaction of the form14 N X JM,J i C JM J ; J C J, − MJ ,J , − J ; J ,J , − J p (2 J + 1)(2 J + 1)(2 J + 1)(2 J + 1) tr (cid:16) [Φ J a , Φ J b ][(Φ † b ) J , (Φ † a ) J ] (cid:17) = 12 N ∞ X l,m,n =0 m + n + l + 1 tr (cid:16) [Φ m + la , Φ nb ][(Φ † b ) n + l , (Φ † a ) m ] (cid:17) . (2.101)In this case the sum over J in the first line is trivial because the conditions on momentasaturate the triangle inequalities, and this fixes J = J + J = J + J . The second line is thenobtained with straightforward shifts and rescalings of the labels.The remaining interactions of the sector all arise from the Yukawa cubic term in the N = 4action, and they generalize Eq. (2.62). Due to the broader field content of the sector withrespect to the previous cases, it is now possible to obtain effective quartic interactions whichsurvive the limit by integrating out three different non-dynamical fields: ψ , ψ or Φ . We start by integrating out the fermion fields ψ , ψ , which works conceptually in the sameway as for the SU(1 , |
1) sector and brings to an effective interaction analog to Eq. (2.73).Since there is an additional SU(2) structure, we find more possible combinations of the fields.The sum over intermediate momenta J can be performed with a shift J → J − in appropriateterms, leading again to a contribution coming from the boundary of summation. We find12 N ∞ X m,n,l =0 tr (cid:16) [(Φ † a ) m , ( ψ b ) m + l ][( ψ b ) † n + l , (Φ a ) n ] (cid:17)p ( m + l + 1)( n + l + 1) − N ∞ X m,n,l =0 s m + 1 n + l + 1 (cid:15) ac (cid:15) bd tr (cid:16) [( ψ † a ) m , (Φ b ) m + l +1 ][( ψ † c ) n + l , (Φ d ) n ] (cid:17) m + n + l + 2 − N ∞ X m,n,l =0 s m + 1 n + l + 1 (cid:15) ac (cid:15) bd tr (cid:16) [(Φ † a ) m + l +1 , ( ψ b ) m ][(Φ † c ) n , ( ψ d ) n + l ] (cid:17) m + n + l + 2 . (2.102)The last interaction comes from integrating out the non-dynamical scalar Φ from the Yukawaterm. This gives rise to a new quartic combination of purely fermionic fields, whose explicitexpression can be worked out by similar manipulations as above, giving12 N ∞ X m,n,l =0 s ( m + 1)( n + 1)( m + l + 1)( n + l + 1) tr (cid:16) { ( ψ a ) m + l , ( ψ b ) n }{ ( ψ † b ) n + l , ( ψ † a ) m } (cid:17) m + n + l + 2 . (2.103)This concludes the treatment of the interacting Hamiltonian of the PSU(1 , |
2) theory. Thefull Hamiltonian of the system in the near-BPS limit g → H ≥ S + Q + Q is H limit = L + ˜ g H int . (2.104)The interacting Hamiltonian is obtained by using the definition (2.94) and collecting all theprevious terms. Remarkably, the final expression can be written in a convenient form showingthat it is manifestly positive definite by means of the propertytr (cid:16) { ψ , ψ } [Φ † , Φ † ] (cid:17) = tr (cid:16) [ ψ , Φ † ][ ψ , Φ † ] (cid:17) − tr (cid:16) [ ψ , Φ † ][ ψ , Φ † ] (cid:17) . (2.105)22hus we obtain H int = 12 N ∞ X l =1 l tr (cid:16) ˆ q † l ˆ q l (cid:17) + 12 N ∞ X l =0 tr (cid:16) ( F ab ) † l ( F ab ) l (cid:17) + 12 N ∞ X l =1 tr (cid:16) ( G ab ) † l ( G ab ) l (cid:17) , (2.106)where we defined ( F ab ) l = ∞ X m =0 [( ψ a ) m + l , (Φ b ) † m ] √ m + l + 1 , ( G ab ) l = 1 √ l ( j ab ) l , (2.107)( j ab ) l = ∞ X m,n =0 s m + 1 n + 1 { ( ψ a ) m , ( ψ b ) n } δ ( m + n + 2 − l ) + [(Φ a ) m , (Φ b ) n ] δ ( m + n + 1 − l ) ! . (2.108)The expression (2.106) is invariant under the global group SU(1 , × U(1) , as can be checkedexplicitly by computing the commutators with the particle number operator ˆ N and withthe charges L , L ± . In addition, it is also invariant under extended supersymmetry, withsupercharges Q = X a =1 , ∞ X n =0 r n + 12 tr (cid:16) ( ψ † a ) n (Φ a ) n +1 + (Φ † a ) n ( ψ a ) n (cid:17) , Q = X a,b =1 , ∞ X n =0 r n + 12 (cid:15) ab tr (cid:16) ( ψ † a ) n (Φ b ) n +1 + (Φ † b ) n ( ψ a ) n (cid:17) . (2.109)satisfying L = {Q , Q † } D = {Q , Q † } D , { H int , Q } D = { H int , Q } D = 0 . (2.110)This can be shown to be true by using the Dirac brackets (2.29) and (2.54) for all the copiesof the fields. The same comments given in Section 2.3 about the non-relativistic nature of themodel are true. In addition, we observe that the broader field content of this near-BPS limitallows for a set of new interactions in the last two lines of Eq. (2.106), where the distributionof momenta between the bosonic and fermionic degrees of freedom is shifted by unity. Thisaspect is related to the fact that the scalars transform under the j = 1 / , , while the fermionic field under the j = 1 representation. We will investigate in moredetails the consequences of this observation in Section 5, where this will play an importantrole to determine the local description of the sector. In Section 2 we found non-relativistic theories that describe the effective dynamics of N = 4SYM near BPS bound, when taking the near-BPS limit (2.13). These theories are classical,as they arise from limits of the classical Hamiltonian of N = 4 SYM on a three-sphere. Inthis section we consider the quantization of the near-BPS theories that we have obtained.In Section 3.1 we quantize the SU(1 , |
1) near-BPS theory and find its full quantummechanical Hamiltonian and the Hilbert space on which it acts. We show that the quantizedtheory includes normal-ordering effects that can be viewed as self-energy corrections. With23hese effects included, we review in Section 3.1 that the quantized SU(1 , |
1) near-BPS theoryis equivalent to SU(1 , |
1) Spin Matrix theory [5]. As we explain in Section 3.2, this meansthat taking the near-BPS limit (2.13) on the level of the classical Hamiltonian of N = 4 SYMon a three-sphere, as we did in Section 2, and then quantizing the resulting near-BPS theory,is equivalent to first quantizing N = 4 SYM on a three-sphere, and then taking the near-BPSlimit (2.13) of the quantum Hamiltonian, which is equivalent to the dilatation operator of N = 4 SYM. Thus, one gets the same result whether one first quantizes, and then takes thenear-BPS limit, or if one first takes the near-BPS limit, and then quantizes.We stress that since one can view the bosonic and fermionic SU(1 ,
1) near-BPS theories astruncations of the SU(1 , |
1) near-BPS theory, the conclusions we draw for the SU(1 , |
1) casewill hold for these cases as well. We furthermore comment on the extension to the PSU(1 , | (1 , | near-BPS theory We perform now the complete quantization procedure for the SU(1 , |
1) near-BPS theory. Thistheory is rich enough to show the appearance of non-trivial contributions from the normalordering of both the bosonic and fermionic terms in the Hamiltonian, and it is also the simplestcase where supersymmetry arises. The procedure can be straightforwardly generalized toinclude the new interactions of the PSU(1 , |
2) sector as no additional subtleties arise.First of all, we replace all the Dirac brackets with (anti)commutators {· , ·} D → i [ · , ·} , (3.1)where we denoted with {} D in the LHS the classical brackets and in the RHS the notationstresses that the symmetry depends from the bosonic or fermionic nature of the fields involved.Then we introduce raising and lowering operators obeying the canonical commutation relations[( a r ) ij , ( a † s ) kl ] = δ il δ kj δ rs , { ( b r ) ij , ( b † s ) kl } = δ il δ kj δ rs , (3.2)where a s ≡ Φ s , a † s ≡ Φ † s are bosonic, and b s ≡ ψ s , b † s ≡ ψ † s are fermionic. These oscilla-tors carry indices i, j for the internal SU( N ) symmetry and an index s corresponding to arepresentation of the spin group SU(1 , | . Using this dictionary, we directly promote the classical result (2.76) to a quantum-mechanicalHamiltonian H qm = tr ∞ X s =0 (cid:18) s + 12 (cid:19) a † s a s + ∞ X s =0 ( s + 1) b † s b s + ˜ g N ∞ X s =1 s ( q tot s ) † q tot s ! + ˜ g N ∞ X s,s ,s =0 p ( s + s + 1)( s + s + 1) tr (cid:16) [ a s , b † s + s ][ b s + s , a † s ] (cid:17) , (3.3)where we defined the quantum version of the charge densities as q l ≡ ∞ X s =0 : [ a † s , a s + l ] : , ˜ q l = ∞ X s =0 √ s + 1 √ s + l + 1 : { b † s , b s + l } : , ˆ q l = q l + ˜ q l . (3.4)At the classical level, the zero mode of the total current ˆ q vanishes due to the Gauss law onthe three-sphere. At the quantum-mechanical level, ˆ q is zero when acting on physical statesˆ q | phys i = 0 . (3.5)24ence, the Hilbert space of the quantum theory corresponds to the states which are singletswith respect to the SU( N ) symmetry. Now we show that normal ordering is responsible forthe appearance of the self-energy corrections. The result for the SU(1 ,
1) bosonic sector wasderived in [4], but here we review and generalize the procedure including also the fermionicpartner.For the bosonic part, the following result can be obtained by using the commutationrelations (3.2) in the explicit evaluation of the normal ordered interaction: ∞ X s =1 s tr (cid:16) q † s q s (cid:17) = ∞ X l =1 s tr (cid:16) : q † s q s : (cid:17) + 2 N ∞ X s =0 h ( s ) tr (cid:16) a † s a s (cid:17) − ∞ X s =0 h ( s ) tr (cid:16) a † s (cid:17) tr( a s ) . (3.6)Here we defined the harmonic numbers as h ( s ) = P sk =1 1 k . An analog computation applied tothe fermionic part of the Hamiltonian gives the similar relation ∞ X s =1 s tr (cid:16) ˜ q † s ˜ q s (cid:17) = ∞ X l =1 s tr (cid:16) : ˜ q † s ˜ q s : (cid:17) + 2 N ∞ X s =0 h ( s + 1) tr (cid:16) b † s b s (cid:17) + 2 ∞ X s =0 h ( s + 1) tr (cid:16) b † s (cid:17) tr( b s ) . (3.7)In this case, the different argument of the harmonic numbers comes from the identity ∞ X l =1 l s + 1 s + l + 1 = h ( s + 1) , (3.8)which in turn arises from the normalization of the fermionic ladder operators. In the SU(1 , | ∞ X s =1 s tr (cid:16) q † s ˜ q s + ˜ q † s q s (cid:17) = ∞ X s =1 s tr (cid:16) : q † s ˜ q s + ˜ q † s q s : (cid:17) . (3.9)To proceed further, we need to work out the implication of the SU( N ) singlet constraint,which implicitly enters the Hamiltonian as the term with s = 0 in the quartic interactionsmediated by the non-dynamical gauge field. For the bosonic case, we need to use the identity ∞ X m =0 h ( m ) tr (cid:16) : [ a † m , a m ] : q (cid:17) = ∞ X m,n =0 h ( m ) tr (cid:16) : h a † m , a m i h a † n , a n i : (cid:17) + 2 N ∞ X m =0 h ( m ) tr (cid:16) a † m a m (cid:17) − ∞ X m =0 h ( m ) tr (cid:16) a † m (cid:17) tr( a m ) , (3.10)while for the fermionic case the analog expression is ∞ X m =0 h ( m + 1) tr (cid:16) : n b † m , b m o : ˜ q (cid:17) = ∞ X m,n =0 h ( m + 1) tr (cid:16) : n b † m , b m o n b † n , b n o : (cid:17) + 2 N ∞ X m =0 h ( m + 1) tr (cid:16) b † m b m (cid:17) + 2 ∞ X m =0 h ( m + 1) tr (cid:16) b † m (cid:17) tr( b m ) . (3.11) Strictly speaking, the singlet constraint involves the total charge density ˆ q and not the single terms q , ˜ q . However the mixed terms do not have normal ordering issues because the bosonic operators commute withthe fermionic ones, hence it is not restrictive to consider only the diagonal terms in the computation of theself-energy corrections.
Commutative diagram representing two different ways to obtain the same quantum theory.One path is to first take the near-BPS limit classically, and then to quantize the theory. That is thepath of this paper performed in Sections 2 and 3.1. The other path is to first quantize N = 4 SYMand then take the near-BPS limit. This corresponds to the Spin Matrix theory limit of [5]. Both pathsyield the same result. No additional self-energy corrections arise instead from the mixed bosonic-fermionic inter-action. The crucial observation is that all the self-energy terms cancel when summing theright-hand side of Eqs. (3.6), (3.7), (3.9) with Eqs. (3.10) and (3.11).The quartic interaction which was mediated by the non-dynamical fermionic field (asexplained in Section 2.3) is already normal-ordered. In this way the quantum Hamiltonian ofthe near-BPS SU(1 , |
1) theory becomes H qm = ∞ X s =0 (cid:18) s + 12 (cid:19) tr (cid:16) a † s a s (cid:17) + ∞ X s =0 ( s + 1) tr (cid:16) b † s b s (cid:17) + ˜ g N ∞ X s =1 s tr (cid:16) : ˆ q † s ˆ q s : (cid:17) + ˜ g N ∞ X s ,s =0 h ( s ) tr (cid:16) : h a † s , a s i h a † s , a s i : (cid:17) + ˜ g N ∞ X s ,s =0 h ( s + 1) tr (cid:16) : n b † s , b s o n b † s , b s o : (cid:17) + ˜ g N ∞ X s ,s =0 h ( s + 1) tr (cid:16) : { b † s , b s } [ a † s , a s ] : (cid:17) + ˜ g N ∞ X s ,s =0 h ( s ) tr (cid:16) : [ a † s , a s ] { b † s , b s } : (cid:17) + ˜ g N ∞ X s,s ,s =0 p ( s + s + 1)( s + s + 1) tr (cid:16) : [ a s , b † s + s ][ b s + s , a † s ] : (cid:17) . (3.12)We will see in the following that this is equivalent to SU(1 , |
1) SMT [5].
Above in Section 2 we have taken near-BPS limits (2.13) of classical N = 4 SYM on a three-sphere, to obtain a classical description of the near-BPS dynamics close to certain BPS bounds.Subsequently we quantized the resulting near-BPS theory, specifically in the SU(1 , |
1) case,to obtain the quantum Hamiltonian (3.12) in Section 3.1. This was done by using a standardnormalordering prescription. In this way we found a quantum Hamiltonian that effectivelydescribes a lower-dimensional theory with non-relativistic symmetries. The route to obtainthis result is illustrated in one of the paths in the diagram of Figure 2.As we shall see in this section, there is another route to the same result, also illustrated inFigure 2. In this case, we start by quantizing N = 4 SYM on a three-sphere. The quantumHamiltonian is then given by the full dilatation operator D of N = 4 SYM on R [10–12]26by the state/operator correspondence). One can then subsequently take the same near-BPSlimit (2.13) as for the classical description. Amazingly, as we show in detail below, this willreveal the exact same quantum theory. Thus, in short, quantizing and taking near-BPS limitcommute with each other.The alternative route, with quantizing first and then taking the near-BPS limit, has beenpreviously explored in [5] and in references therein. In these works, the near-BPS limit isknown as the Spin Matrix theory (SMT) limit and the resulting quantum theory as SpinMatrix theory (SMT). Thus, we show in this paper a different route to obtain SMT.That quantization and near-BPS limit commutes, as illustrated in Figure 2, is not apriori evident. The commutativity of the limits is particularly non-trivial for non-relativistictheories: in fact it is known that procedures like the c → ∞ limit or null reduction do notcommute a priori with the quantization of the theory, or with other generic limits that onecan perform in such systems. For our near-BPS limits, however, we show that the diagram inFig. 2 is commutative, i.e. the two prescriptions lead to the same result. A posteriori, thismatching justifies the prescription given for the quantization of the classical result comingfrom the sphere reduction.To exhibit the connection to the SMT/near-BPS limits of the full dilatation operator D of N = 4 SYM we focus on the SU(1 , |
1) case for which we found the quantum Hamiltonian(3.12). The focus here is on the interacting part which should be compared to the SU(1 , | D . Writing H qm = ∞ X s =0 (cid:18) s + 12 (cid:19) tr (cid:16) a † s a s (cid:17) + ∞ X s =0 ( s + 1) tr (cid:16) b † s b s (cid:17) + ˜ g H qm , int (3.13)we are interested in the quantum interacting Hamiltonian H qm , int . It is convenient to write itin terms of renormalized four-point vertices, H qm , int = 14 N ∞ X s =0 s X s ,s =0 tr (cid:16) : [ a † s , a s ][ a † s − s , a s − s ] : (cid:17) (cid:18) δ s ,s ( h ( s ) + h ( s − s )) − − δ s ,s | s − s | (cid:19) + ˜ g N ∞ X s =0 s X s ,s =0 s ( s + 1)( s − s + 1)( s + 1)( s − s + 1) tr (cid:16) : { b † s , b s }{ b † s − s , b s − s } : (cid:17) × (cid:18) δ s ,s ( h ( s + 1) + h ( s − s + 1)) − − δ s ,s | s − s | (cid:19) + ˜ g N ∞ X s =0 s X s ,s =0 s s − s + 1 s − s + 1 tr (cid:16) : [ a † s , a s ] { b † s − s , b s − s } : (cid:17) (cid:18) δ s ,s h ( s ) − − δ s ,s | s − s | (cid:19) + ˜ g N ∞ X s =0 s X s ,s =0 s s + 1 s + 1 tr (cid:16) : { b † s , b s } [ a † s − s , a s − s ] : (cid:17) (cid:18) δ s ,s h ( s + 1) − − δ s ,s | s − s | (cid:19) + ˜ g N ∞ X s,s ,s =0 p ( s + s + 1)( s + s + 1) tr (cid:16) : [ a s , b † s + s ][ b s + s , a † s ] : (cid:17) . (3.14)The terms in the right parenthesis for each vertex correspond exactly to the one-loop dilatationoperator in the SU(1 , |
1) sector [11]. Thus, we have shown the commutativity in the diagramof Figure 2. This means that the quantum Hamiltonian (3.12) indeed is that of SU(1 , | N = 4 SYM, is quite involved. One has to compute divergent Feynmandiagrams for N = 4 SYM and perform dimensional regularization for two-point functions.Instead, we have found the same result from a classical computation, i.e. the near-BPSlimit of the sphere reduction of the action on R × S , along with a simple normal-orderingprescription to obtain the Hamiltonian at the quantum level. Generalization to the PSU (1 , | sector Here we comment on the generalization of the result to the PSU(1 , |
2) sector. In this case,the bosonic and fermionic fields are both supplemented by an additional SU(2) index due tothe residual R-symmetry of the system. We then define the ladder operators as( a a ) s ≡ (Φ a ) s , ( a † a ) s ≡ (Φ † a ) s , ( b a ) s ≡ ( ψ a ) s , ( b † a ) s ≡ ( ψ † a ) s . (3.15)The prescription to quantize the Hamiltonian coming from the sphere reduction is still todirectly promote the result at quantum level, without further changes. This implies that theinteracting part is given by H qm , int = 12 N ∞ X l =1 l tr (cid:16) ˆ q † l ˆ q l (cid:17) + 12 N ∞ X l =0 tr (cid:16) ( F ab ) † l ( F ab ) l (cid:17) + 12 N ∞ X l =1 tr (cid:16) ( G ab ) † l ( G ab ) l (cid:17) , (3.16)where now we have q l ≡ X a =1 , ∞ X n =0 : [( a † a ) n , ( a a ) n + l ] : , ˜ q l ≡ X a =1 , ∞ X n =0 √ n + 1 √ n + l + 1 : { ( b † a ) n , ( b a ) n + l } : , (3.17)( F ab ) l = ∞ X m =0 [( b a ) m + l , ( a b ) † m ] √ m + l + 1 , ( G ab ) l = 1 √ l ( j ab ) l , (3.18)( j ab ) l = ∞ X m,n =0 s m + 1 n + 1 { ( b a ) m , ( b b ) n } δ ( m + n + 2 − l ) + [( a a ) m , ( a b ) n ] δ ( m + n + 1 − l ) ! . (3.19)Working out the SU( N ) singlet condition and writing all the expressions in terms of normalordered quantities allows to recast the result in a form where the vertices are renormalizedin the same way as computed from the one-loop corrections to the dilatation operator in thissector. The procedure is completely analog to the SU(1 , |
1) case, and we simply need tocomplement the result with the additional SU(2) structure.
The spin group of the SMT Hamiltonian in the SU(1 , |
1) limit is supersymmetric, i.e. itadmits the existence of a complex supercharge relating the bosonic and fermionic dynamicaldegrees of freedom surviving the near-BPS limit of N = 4 SYM. It is then reasonable to expectthat there exists a suitable superspace formulation which makes this invariance manifest andallows to reproduce the field content and the Hamiltonian in terms of superfields.28ndeed, we now show in detail that this is possible. We stress that while we will give asemi-local description of this model in section 5, it should be considered as a complementaryway to describe the system, but not as a necessary step. In fact, all the expressions that weare going to introduce in this section can be considered independently as a way to obtain theclassical Hamiltonian (2.75).Following the discussion of Section 2.3, it is convenient to use L as the free part of theHamiltonian; this reads L = Z dt ∞ X s =0 tr (cid:18)(cid:18) s + 12 (cid:19) Φ † s Φ s + ( s + 1) ψ † s ψ s (cid:19) . (4.1)The eigenvalues are explicitly given by s + 1 − R, with R = ( ,
0) being the U (1) R charge ofbosons and fermions , respectively.Since the sector contains only a single complex supercharge, a corresponding superspaceformulation accordingly requires the introduction of a single complex Grassmannian coordi-nate ( θ, θ † ) . Moreover, the requirement that the anticommutator of supercharges closes on L fixes their expressions to be Q = ∂∂θ + 12 θ † ( s + 1 − R ) , Q † = ∂∂θ † + 12 θ ( s + 1 − R ) , (4.2)which indeed satisfy {Q , Q † } = s + 1 − R = L . (4.3)The most general superfield that we can define in a superspace with one complex Grassmanncoordinate is given by X s ( t, θ, θ † ) = A s ( t ) + θB s ( t ) + θ † C s ( t ) + θθ † D s ( t ) . (4.4)The component modes appearing in the definition of the superfield can have a priori bothbosonic or fermionic statistics. In particular, in two dimensions both choices are allowed.We will distinguish the two possibilities by calling the superfields either bosonic or fermionicdepending from the behaviour of the lowest component. In fact, fixing the statistics of A s ( t )is sufficient to fix the statistics of all the other component fields in the expansion.Given the general expression of the superfield, it turns out that the number of componentfields in the multiplet is too big, and we need some constraints in order to find an irreduciblerepresentation. This task can be achieved by defining the covariant derivatives D = i ∂∂θ − i θ † ( s + 1 − R ) , D † = − i ∂∂θ † + i θ ( s + 1 − R ) , (4.5)which satisfy the following commutation relations: { D, Q} = { D † , Q † } = { D, Q † } = { D † , Q} = 0 , { D, D † } = − L . (4.6)In this way we define the notion of chiral Σ s and anti-chiral Σ † s superfields by requiring theconditions D † Σ s = 0 , D Σ † s = 0 . (4.7) See Appendix D for more details on the R-charge and the generators in the oscillator representation.
29e will show that the only matter field needed to build the Hamiltonian in superfield languageis a chiral fermionic superfield Ψ plus its hermitian conjugate Ψ † . For this reason, we directlyconsider the case where the bottom component of the supermultiplet is a complex fermionand we impose the (anti)chirality constraints to getΨ s ( t, θ, θ † ) = ψ s ( t ) √ s +1 + θ Φ s ( t ) − θθ † √ s + 1 ψ s ( t ) , (4.8)Ψ † s ( t, θ, θ † ) = ψ † s ( t ) √ s +1 + θ † Φ † s ( t ) − θθ † √ s + 1 ψ † s ( t ) . (4.9)Notice that the particular normalization of the fermionic components reflects the definition ofthe charge densities, see Eqs. (2.42) and (2.69). This will play an important role to determinethe correct form of the interactions. The constrained superfield gives an irreducible mattersupermultiplet, since it only contains a single complex scalar and the fermionic partner, whichare the surviving degrees of freedom of the near-BPS limit. No auxiliary fields are needed.The supersymmetry transformations of all the modes can be found by computing δ Ψ s = (cid:16) (cid:15) Q + (cid:15) † Q † (cid:17) Ψ s , (4.10)and then projecting the result on the various components. We find that the free Hamiltonian L can be written as L = − Z dt Z dθdθ † ∞ X s =0 tr (cid:16) Ψ † s ( s + 1 − R )Ψ s (cid:17) . (4.11)Working out the rules of Berezin integration, this is easily shown to correspond in componentformalism to Eq. (4.1).Now we move to the interacting part of the Hamiltonian. Having at disposal the fermionicsuperfield containing all the dynamical fields of the theory, in principle we can build higher-order terms with appropriate combinations of the superfield. However, it turns out that thechoice of the fermion as the lowest component of the supermultiplet and the Grassmanniannature of the superspace coordinates are responsible for the identitiesΨ s = 0 , (Ψ † s ) = 0 , (4.12)which are a natural supersymmetric generalization of the concept of Grassmann variable.In particular, this fact rules out the construction of a superpotential, i.e. an expressionholomorphic in the superfields, which is the natural candidate for renormalizable interactionsin standard relativistic theories. While this fact forbids to build the SU(1 , |
1) Hamiltonianonly in terms of the fermionic superfield, on the other hand it shows that another kind ofsupermultiplet is required to specify the theory, and in fact we will need to add a bosonic(anti)chiral superfield. The necessity to integrate in a new field also arises at the level ofcomponents, as we will show. This is justified by the fact that the theory still containsremnants of the original gauge symmetry of the N = 4 SYM action, which in the near-BPSlimit are non-dynamical and mediate the interactions via the currents associated to the matterfields.Using the component formalism, we define the modes of the current to be j l = ∞ X s =0 s s + 1 s + l + 1 n ψ † s , ψ s + l o + h Φ † s , Φ s + l i! . (4.13)30otice that this definition is exactly the total current ˆ q l = q l + ˜ q l written in terms of bosonicand fermionic currents of Eqs. (2.42) and (2.69). Now we introduce a gauge contribution tothe Hamiltonian given by a kinetic term for a complex scalar mode A s ( t ) and the minimalcoupling between such field and the current: H ⊃ Z dt − ∞ X s =0 s tr (cid:16) A † s A s (cid:17) + ˜ g ∞ X s =0 tr (cid:16) A † s j s + A s j † s (cid:17)! . (4.14)The equation of motion for the constrained gauge field A s in Fourier space is sA s − ˜ gj s = 0 , (4.15)and after integrating it out, the term added to the Hamiltonian becomes Z dt ∞ X s =1 s tr (cid:16) j † s j s (cid:17) . (4.16)This shows that by integrating in an auxiliary gauge mediator we get precisely this termentering the Hamiltonian. The quartic interaction between two scalars and two fermions inEq. (2.76) can be obtained in component formalism by simply combining the fields as H ⊃ Z dt ∞ X s ,s ,l =0 p ( s + l + 1)( s + l + 1) tr (cid:16) [Φ s , ψ † s + l ][ ψ s + l , Φ † s ] (cid:17) ! . (4.17)How is possible to obtain this term from the superfield perspective if we cannot buildholomorphic combinations of fermionic superfields? It turns out that the supersymmetrizationof the gauge mediator will solve the problem at once, accounting for both the term containingthe currents and the quartic mixed interaction.In fact, we define the following bosonic (anti)chiral superfield A s ( t, θ, θ † ) = A s ( t ) + θ λ s ( t ) √ s +1 − θθ † sA s ( t ) , (4.18) A † s ( t, θ, θ † ) = A † s ( t ) − θ † λ † s ( t ) √ s +1 − θθ † sA † s ( t ) . (4.19)Since the theory is supersymmetric, we had to introduce in the definition a complex fermion λ s , which we will interpret as a residual gaugino mediating another interaction. In this way,we can write the complete Hamiltonian of the sector as H = Z dt Z dθdθ † tr ( ∞ X s =0 (cid:16) A † s A s − Ψ † s ( s + 1 − R ) Ψ s (cid:17) +˜ g ∞ X s ,s =0 Ψ † s h A † s − s + R − , Ψ s i + ˜ g ∞ X s ,s =0 Ψ † s h A s − s + R − , Ψ s i . (4.20)Although this is not manifest, the terms in the second line can be interpreted as a covariantderivative i D x written in momentum space, as we will see with a local formulation in section We change here the notation of the current as j l instead of ˆ q l to avoid confusion with supersymmetrycharges and to stress that it plays the role of a current in a QFT coupling to a mediator gauge field. H = Z dt ∞ X s =0 tr (cid:18) s + 12 (cid:19) Φ † s Φ s + ( s + 1) ψ † s ψ s − ∞ X s =0 sA † s A s +˜ g ∞ X s =0 (cid:16) A † s j s + A s j † s (cid:17) − ˜ g ∞ X s,s ,s =0 λ s [ ψ s , Φ † s ] + ˜ g ∞ X s,s ,s =0 λ † s [Φ s , ψ † s ] . (4.21)The first line contains all the kinetic terms, while the second line all the couplings withthe currents. The remarkable fact is that the gaugino λ s is not dynamical, and can beeasily integrated out giving a quartic interaction which corresponds exactly to Eq. (4.17).Then we see how the superfield formulation solves the problem: the fermionic partner of theremnant gauge field allows to build a term using minimal coupling without resorting to any(anti)holomorphic superpotential. At this point we observe that the field A s is also non-dynamical, and following the step in Eq. (4.15) we integrate out this field to get the quarticinteraction (4.16). In Section 2 we have presented non-relativistic near-BPS theories that arise from limits ofclassical N = 4 SYM on a three-sphere. In Section 3 we have quantized these theoriesemploying a normal-ordering prescription. The quantized theories are the Spin Matrix theories(SMTs) considered in [5] and references therein, that also can be obtained directly from limitsof quantized N = 4 SYM.In this section we find local formulations of the quantized near-BPS theories/SMTs. Ourmain focus is on SU(1 , |
1) SMT, but we also comment on the other cases with SU(1 , ,
1) sector as a simplesetting to introduce the procedure, and we finally comment on the PSU(1 , |
2) case, beingthe one with richest structure.
In all of the four cases that we consider in this paper we have a SU(1 ,
1) subalgebra of thebosonic part of the algebra. This SU(1 ,
1) is the non-compact part of the algebra, and theSU(1 ,
1) representations that we have are infinite dimensional. For this reason, one can find alocal representation of the states that we have with respect to their SU(1 ,
1) representations.We shall do this below for the SU(1 , |
1) SMT by considering just the free Hamiltonian.Subsequently we include the interactions: in Section 5.2 we start from the SU(1 ,
1) bosonicSMT, we then consider in Section 5.3 the full SU(1 , |
1) SMT and we finally comment on thePSU(1 , |
2) case in Section 5.4.In the SU(1 , |
1) SMT limit of N = 4 SYM the surviving states are | d n Z i and | d n χ i with n ≥ ,
1) representation ofthese states. SU(1 ,
1) has three generators L , L + and L − with algebra[ L , L ± ] = ± L ± , [ L − , L + ] = 2 L . (5.1)32cting with the three generators on the surviving states one finds L | d n Z i = (cid:16) n + (cid:17) | d n Z i , L | d n χ i = ( n + 1) | d n χ i ,L + | d n Z i = ( n + 1) | d n +11 Z i , L + | d n χ i = p ( n + 1)( n + 2) | d n +11 χ i ,L − | d n Z i = n | d n − Z i , L − | d n χ i = p n ( n + 1) | d n − χ i , (5.2)with n ≥
0. One can compare this to a general spin j representation of SU(1 , L | j, j + n i = ( j + n ) | j, j + n i ,L + | j, j + n i = q ( n + 1)( n + 2 j ) | j, j + n + 1 i ,L − | j, j + n i = q n ( n + 2 j − | j, j + n − i . (5.3)This shows that the bosonic states | d n Z i are in the j = 1 / | d n χ i are in the j = 1 representation of SU(1 , L and L ± in terms of differential operators on alocal field. Since we have only one quantum number m in the j = 1 / j = 1 representationsit should be a one-dimensional spatial direction. Moreover, since it’s quantized, one shouldput it on a circle. Hence we introduce the spatial coordinate x , periodic with period 2 π , toparametrize this circle.Consider first the bosonic states | d n Z i . We introduce the bosonic complex fieldΦ( t, x ) = ∞ X n =0 Φ n ( t ) e i ( n + ) x . (5.4)Note that we are in the Heisenberg picture. Identifying L = n + one sees that L = − i∂ x .This means that as a quantum operator Φ † n acting on the vacuum creates the state | d n Z i .Note that Φ is antiperiodic on the circle due to the half-integer momentum on the circle. Aswe shall see below, Φ shares some features with β - γ ghost fields, and thus has a mixture ofbosonic and fermionic characteristics. A consistent representation on Φ( t, x ) of the SU(1 , L = − i∂ x , L ± = e ± i ( x − t ) ( − i∂ x ± R ) , (5.5)since this reproduces the algebra (5.1). Here R is the U(1) charge which is R = for bosonsand R = 0 for fermions. The time-dependence in (5.5) will be addressed below. An importantquestion is the normalization of the mode Φ n . The states | d n Z i = | , + n i are normalized.Hence one can read off this normalization from the action with L ± in (5.2). This shows thatΦ n is normalized and we have [Φ m , Φ † n ] = iδ mn . (5.6)One can now evaluate the equal-time commutators of Φ( t, x ) giving the result[Φ( t, x ) , Φ( t, x )] = 0 , [Φ( t, x ) , Φ( t, x ) † ] = iS ( x − x ) , (5.7)where S j ( x ) = ∞ X n =0 e i ( n + j ) x (5.8)This points to the fact that Φ( t, x ) does not have the standard behavior of a local field. Inparticular, even if one can find a quantum state that is an eigenstate of the momentum alongthe circle, one cannot find a quantum state which is an eigenstate of the position along x .33urning to the fermionic states | d n χ i we introduce the Grassmann-valued complex field ψ ( t, x ) = ∞ X n =0 √ n + 1 ψ n ( t ) e i ( n +1) x . (5.9)Identifying L = n + 1 we have again the representation (5.5) of the SU(1 ,
1) algebra on ψ ( t, x ). The field ψ ( t, x ) is periodic in the x direction. Upon quantization, the mode ψ † n acting on the vacuum gives the state | d n χ i . Again, one should check the normalization of ψ n with the action of L ± in (5.1) and (5.2). With the 1 / √ n + 1 factor in (5.9) one gets that ψ n is normalized, hence in the quantized theory we have { ψ m , ψ † n } = iδ mn . (5.10)With this, the equal-time anti-commutators of ψ ( t, x ) are { ψ ( t, x ) , ψ ( t, x ) } = 0 , { ψ ( t, x ) , i∂ x ψ ( t, x ) † } = iS ( x − x ) , (5.11)Again, this is not a standard anti-commutator for a local fermionic quantum field. (1 , SMT
We start discussing the basic procedure to build a QFT description for the simplest near-BPS limit, i.e. the SU(1 ,
1) bosonic sector. The main task is to reproduce the interactingHamiltonian in Eq. (2.45), which is given in momentum space, in terms of a local field theorycontaining the complex scalar field (5.4) satisfying the equal time commutator (5.7). Thepresence of the singlet constraint in the Hamiltonian implies that the SU( N ) remains gauged.Moreover, we need to integrate in an additional auxiliary field in order to reproduce theinteractions. We can interpret this step as the position space version of the mediation givenby the non-dynamical gauge field in the sphere reduction procedure described in Section 2.Consider the following (1+1)-dimensional field theory on a circle of unit radius parametrizedby the spatial coordinate x with periodic indentification x ∼ x + 2 πS = Z dtdx tr (cid:16) i Φ † ( ∂ + ∂ x )Φ + iA † ∂ x A + ˜ g (cid:16) A † j + Aj † (cid:17)(cid:17) , (5.12)where j ( t, x ) is the charge density associated to the SU( N ) symmetry defined by j ( t, x ) ≡ [Φ † ( t, x ) , Φ( t, x )] . (5.13)We show that the previous local action gives rise to Eq. (2.45) with an appopriate decompo-sition of the auxiliary field in momentum space. We require A ( t, x ) = ∞ X n =0 A n ( t ) e inx . (5.14)Combining this expansion with the scalar one in Eq. (5.4) we obtain S = ∞ X n =0 Z dt tr (cid:18) i Φ † n ∂ t Φ n + (cid:18) n + 12 (cid:19) Φ † n Φ n + nA † n A n + ˜ g (cid:16) A † n j n + A n j † n (cid:17)(cid:19) , (5.15)34here the modes of the charge density j n ( t ) are given by Eq. (2.42). Since A ( t, x ) is non-dynamical, its equations of motion give rise to the constraint nA n ( t ) + ˜ gj n ( t ) = 0 . (5.16)For n = 0 , this coincides with the SU( N ) singlet constraint j = 0 . When n > , the constraintcan be solved and inserted into the action to get S = Z dt tr " ∞ X n =0 i Φ † n ∂ t Φ n + ∞ X n =0 (cid:18) n + 12 (cid:19) Φ † n Φ n − ˜ g ∞ X n =1 n j † n j n + ˜ g ( A + A † ) j . (5.17)The corresponding Hamiltonian is easily derived via the Legendre transform and correspondsexactly to Eq. (2.45).We conclude the analysis of the SU(1 ,
1) bosonic limit with some comments on the action(5.12). The form of the kinetic term is unusual, being linear in both the time and spacederivatives. In the standard relativistic case the Klein-Gordon operator is quadratic, while inthe Schroedinger-invariant case the action is linear in the time derivative, but quadratic alongthe spatial directions. Instead, this kinetic term corresponds to an ultra-relativistic dispersionrelation between energy and momentum E = P, typical of Carrollian theories [29]. In this case,however, there is the non-trivial constraint P > , which makes the theory non-relativistic.From this perspective, we see that the momentum constraint and the non-standard Diracbrackets become necessary to get a non-relativistic interpretation of the result.Finally, we remark that the scalar must necessarily be complex, otherwise the kinetic termwould be a total derivative. In this connection, it is amusing to note that upon introducingtwo real scalar fields ( β, γ ) as Φ = β + iγ , (5.18)the kinetic term L = i Φ † ( ∂ + ∂ x )Φ of the action (5.12) becomes L = − β ( ∂ + ∂ x ) γ . (5.19)This shows that the bosonic part of the action can be viewed as a β - γ CFT, which is a theorywith negative central charge. Exploring this intriguing connection further is a matter forfuture work. (1 , | SMT
We extend the QFT description of the Section 5.2 to include to the SU(1 , | , which containsa fermionic partner for the scalar field. In particular, we present the result in a manifestlysupersymmetric way by giving the position space version of the superspace formulation intro-duced in Section 4, and then we will comment on the result in terms of component fields.Having identified the free part of the Hamiltonian with L = − i∂ x , we need to searchfor a representation of the supercharge such that {Q , Q † } = − i∂ x . The presence of a singlecomplex supercharge implies that superspace is composed by one complex Grassmann variable θ, a common feature with the momentum space description. It is simple to check that thefollowing representation satisfies the correct anticommutator: Q = ∂∂θ − i θ † ∂ x , Q † = ∂∂θ † − i θ∂ x . (5.20)35onsistency between the left and right multiplication in defining superspace implies that wecan define the supersymmetric covariant derivatives as D = i ∂∂θ − θ † ∂ x , D † = − i ∂∂θ † + 12 θ∂ x , (5.21)satisfying the commutators { D, Q} = { D † , Q † } = { D, Q † } = { D † , Q} = 0 , { D, D † } = i∂ x = − L . (5.22)These expressions correspond in momentum space to Eq. (4.2) and (4.5).The annihilation under the action of the covariant derivatives of a generic superfield allowsto define (anti)chiral superfields. We direcly introduce the quantities that are sufficient tobuild an action in superspace formalism: the (anti)chiral fermionic superfields containing thedynamical modes of the SU(1 , |
1) sectorΨ( t, x, θ, θ † ) = ψ ( t, x ) + θ Φ( t, x ) + i θθ † ∂ x ψ ( t, x ) , (5.23)Ψ † ( t, x, θ, θ † ) = ψ † ( t, x ) + θ † Φ † ( t, x ) − i θθ † ∂ x ψ † ( t, x ) , (5.24)and the bosonic (anti)chiral superfield containing the auxiliary fields A ( t, x, θ, θ † ) = A ( t, x ) + θλ ( t, x ) + i θθ † ∂ x A ( t, x ) , (5.25) A † ( t, x, θ, θ † ) = A † ( t, x ) − θ † λ † ( t, x ) − i θθ † ∂ x A † ( t, x ) . (5.26)When expanding in modes the component fields, we obtain Eq. (4.8), (4.9), (4.18) and (4.19).The gauge superfield (5.25) is composed by fields that we will call a gauge field A ( t, x )and a gaugino λ ( t, x ) in the sense that they play the role of mediators of other interactions,and they are remnants of the original gauge invariance of the N = 4 SYM action beforeimposing Coulomb gauge and performing the sphere reduction. We can further push on thisinterpretation by defining derivatives covariant with respect to the gauge superfield D ≡ ∂ , D x ≡ ∂ x − i ˜ g A − i ˜ g A † . (5.27)When applied on the fermionic superfield, it acts as D x Ψ ≡ ∂ x Ψ − i ˜ g [ A , Ψ] − i ˜ g [ A † , Ψ] , (5.28)where in component formalism the brackets are commutators or anticommutators dependingfrom the statistics of the specific field they are acting on.In this way we obtain a compact expression for the action describing the effective fieldtheory of the SU(1 , |
1) sector S = Z dtdx Z dθ † dθ tr (cid:16) − i Ψ † ( D + D x )Ψ + A † A (cid:17) . (5.29)This proposal is very natural: the matter part is a simple generalization of the two-dimensionalDirac action, with the Dirac spinor replaced by a fermionic superfield. The coupling with theauxiliary field is also straightforward: there is a minimal coupling via the introduction of acovariant derivative containing the real part of the gauge superfield, while the kinetic termis standard for a chiral bosonic superfield. Notice that while in standard cases, e.g. for36he relativistic N = 1 chiral superfield in (3+1)-dimensions, a kinetic term of kind A † A isdynamical, here the specific expansion in superspace (5.25) shows that no time derivativeappears, i.e. the gauge field and the gaugino are non-dynamical. We notice that the set ofinteractions built in this way are quite general, since the Grassmannian nature of the fermionicsuperfield implies Ψ = 0 , (Ψ † ) = 0 , (5.30)so that higher-order (anti)holomorphic terms are forbidden.We further comment on the supersymmetry invariance of the action (5.29). The interactingpart of the action is manifestly invariant under supersymmetry because it is built only usingthe superfield formulation, and it is the non-trivial content of the SU(1 , |
1) sector. Onthe other hand, the kinetic term is not supersymmetric invariant: it is given by L and isdefined using a derivative which is covariant with respect to the gauge superfield (5.25), butnot under supersymmetry. Since the free Hamiltonian given by H in Eq. (2.55) is insteadsupersymmetric invariant and it differs from L by the conserved charge ˆ N in (2.58), it is easyto obtain a manifestly supersymmetric kinetic term by simply adding a mass shift of 1/2 inthe differential operator − i ( D + D x ) . It is instructive to decompose the action (5.29) in component fields. Since it turns outthat the gaugino λ ( t, x ) appears simply as a Lagrange multiplier, we immediately solve thecorresponding constraint to integrate it out. We find S = Z dtdx tr n i Φ † ( ∂ + ∂ x )Φ − ∂ x ψ † ( ∂ + ∂ x ) ψ + iA † ∂ x A + ˜ gAj + ˜ gA † j † + g [Φ , ψ † ][ ψ, Φ † ] o , (5.31)where the scalar field is defined in Eq. (5.4), the fermionic field in (5.9), the gauge field in(5.14) and the current is now given by j ( t, x ) = i { ∂ x ψ † , ψ } + [Φ † , Φ] . (5.32)When expanding it in momentum space, we obtain the charge density ˆ q n = q n + ˜ q n definedfrom Eq. (2.42) and (2.69). Putting the decomposition of all the fields in momentum spaceinside the component field action (5.31), we get the Legendre transform of the Hamiltonian(2.75), with interactions (2.76).Looking at the fermionic kinetic term in the action (5.31), we notice that while thequadratic part in the spatial derivatives is standard e.g. in Schroedinger-invariant theories,instead the product of one time and one spatial derivative is peculiar. Notice that had wetaken the fermionic field to be real, the kinetic term would have been a total derivative. Notefurther that the structure of the fermionic kinetic term is in complete agreement with that ofa complex chiral boson (see e.g. [30]), only that the field is Grassmann valued. Again, thisunveils a curious correspondence with ghost fields with nonstandard statistics.It is also interesting to observe that we obtain a natural superfield description of the modelwith action (5.29) by defining a gauge superfield, which requires the inclusion of a fermionicpartner for the gauge field. However λ ( t, x ) turns out to be completely auxiliary, and in factit is not necessary to introduce it when considering a component field formulation. In thissense, it plays the same role of the auxiliary field F entering the relativistic N = 1 bosonicchiral superfield in 3+1 dimensions. 37 .4 Local formulation of PSU (1 , | SMT
It is straightforward to extend the QFT description of Section 5.3 in order to obtain thefull PSU(1 , |
2) sector. We work in component field formulation and require the followingdecomposition of the fields:Φ a ( t, x ) = P ∞ n =0 (Φ a ) n ( t ) e i ( n + ) x , ψ a ( t, x ) = P ∞ n =0 1 √ n +1 ( ψ a ) n ( t ) e i ( n +1) x , (5.33) A ( t, x ) = P ∞ n =0 A n ( t ) e inx , B ab ( t, x ) = P ∞ n =0 ( B ab ) n ( t ) e inx . (5.34)In addition to the doublet structure of bosons and fermions under SU(2) , we introducedanother bosonic field B ab ( t, x ) which will mediate the interactions; the difference with A ( t, x )is that it will give rise to single trace structures, while the latter will contribute to doubletrace interactions.We then consider the total action S = Z dtdx tr n i Φ † a ( ∂ + ∂ x ) Φ a − ∂ x ψ † a ( ∂ + ∂ x ) ψ a − iA † ∂ x A − iB † ab ∂ x B ab − ˜ gA † j − ˜ gAj † − ˜ gB ab j † ab − ˜ gj ab B † ab − ˜ g | [Φ a , ψ † a ] | o , (5.35)with currents j ( x ) = i { ∂ x ψ † a ( x ) , ψ a ( x ) } + [Φ † a , Φ a ( x )] , j ab ( x ) = − i { ∂ x ψ a ( x ) , ψ b ( x ) } + [Φ a ( x ) , Φ b ( x )] . (5.36)The matching of the kinetic terms in (2.104) with the double trace interactions in (2.106) isstraightforward and works as in Section 5.3. We briefly show how the matching of the singletrace structure works: the current j ab ( t, x ) in Fourier space reads( j ab ) s = ∞ X m,n =0 s m + 1 n + 1 { ( ψ a ) m , ( ψ b ) n } δ ( m + n + 2 − s ) + [(Φ a ) m , (Φ b ) n ] δ ( m + n + 1 − s ) ! . (5.37)This is exactly the expression (2.108). Notice that the mode expansion for the dynamicalbosonic and fermionic fields is shifted by R, which takes the values , j = , ,
1) group. The equations ofmotion for the non-dynamical field B ab in Fourier space are s ( B ab ) s − ˜ g ( j ab ) s = 0 , (5.38)and integrating out this field we get the interaction ∞ X s =1 s tr h ( j † ab ) s ( j ab ) s i , (5.39)which is exactly Eq. (2.106). In this paper we have shown how to take the near-BPS limits (1.2) directly of the classical for-mulation of N = 4 SYM on a three-sphere, following [4]. The BPS bounds we considered were38ll of the form (1.1) giving a surviving SU(1 ,
1) global symmetry along with a U(1)-symmetrycorresponding to conservation of the number operator. In the SU(1 , |
1) and PSU(1 , | ,
1) symmetry is a subgroup of a larger symmetry. The techniquesused for taking the limits include the spherical reduction of N = 4 SYM on a three-sphere,following [4], as well as integrating out non-dynamical fields that in some cases contribute tothe interaction of the surviving modes.We have shown explicitly how to quantize the near-BPS theories, and shown that theresult is equivalent to taking the near-BPS limit directly of the quantized N = 4 SYM. Thismeans the quantized near-BPS theories corresponds to the Spin Matrix theories [5].Finally, we found a superfield formulation of SU(1 , |
1) in momentum space, and we haveexplored a way to represent the near-BPS/Spin Matrix theories as local non-relativistic quan-tum field theories. This has revealed interesting and surprisingly elegant structures for theinteractions, in particular in the SU(1 , |
1) case and its two SU(1 ,
1) subcases. The quantumfields we found are not fully local and have ghost-like features in that the bosonic fields havefermionic features, and the fermionic fields have bosonic features.As explained in the introduction, near-BPS/Spin Matrix theories are interesting in viewof their possible realization of the holographic principle, and, related to this, also as a possibleway to access new regimes in the AdS/CFT correspondence that have yet to be explored.For this reason, it is highly interesting to observe that the form of the interactions in theSU(1 , |
1) theory makes it possible to access the strong coupling. Indeed, we find a classicalHamiltonian of the form (see Section 2.3) H limit = L + ˜ g N ∞ X l =1 l tr (cid:16) ˆ q † l ˆ q l (cid:17) + ∞ X l =0 tr (cid:16) F † l F l (cid:17)! . (6.1)We see that since the terms in the interaction are positive definite, we notice that in thestrong coupling limit ˜ g → ∞ the leading contribution is given by ˆ q l +1 = F l = 0 for l ≥ ,
1) case.A similar argument can be applied to the Hamiltonian of the PSU(1 , |
2) sector H limit = L + ˜ g N " ∞ X l =1 l tr (cid:16) ˆ q † l ˆ q l (cid:17) + ∞ X l =0 tr (cid:16) ( F ab ) † l ( F ab ) l (cid:17) + ∞ X l =1 tr (cid:16) ( G ab ) † l ( G ab ) l (cid:17) , (6.2)suggesting that the strong coupling limit ˜ g → ∞ corresponds to a leading contribution whereˆ q l +1 = ( F ab ) l = ( G ab ) l +1 = 0 for l ≥ . It will be illuminating to understand better theSU(1 , | ,
1) case onefinds a dual U(1)-Galilean geometry which is basically R times a cigar-geometry, where R isthe time-direction. It will be vital to explore this further, also in view of the non-standardfeatures of the SU(1 ,
1) theory formulated as local quantum field theories. Moreover, beyondthe planar limit, the BPS bounds (1.1) examined in this paper are related to limits of blackholes in AdS × S with vanishing entropy. There is also the intriguing possibility that one39an observe the emergence of dual D-branes, in the form of Giant Gravitons, similarly to whatwas found in [14].We found an interesting connection to the β - γ ghost CFT for the kinetic term of the scalarfields, see Eq. (5.19). This would be interesting to explore further as it possibly could provideanother view point on the P > ,
1) may allow for a more natural field theoretic formulationof our near-BPS theories. We note that SU(1 ,
1) representations are also realized on AdS and on the hyperbolic plane.Finally, we would like to advertise our companion paper [31] in which we explore a classof near-BPS limits with a SU(1 ,
2) global symmetry present. Among these near-BPS theoriesare the theory with PSU(1 , |
3) symmetry that captures the behavior of N = 4 SYM nearthe BPS bound E ≥ S + S + J + J + J , a bound saturated by the supersymmetric blackhole in AdS × S [32]. Acknowledgements
We thank Yang Lei for many interesting discussions and useful comments on the draft ofthis paper. We acknowledge support from the Independent Research Fund Denmark grantnumber DFF-6108-00340 “Towards a deeper understanding of black holes with non-relativisticholography”.
A Spherical harmonics on S In this Appendix we review the decomposition of fields on R × S into a basis of sphericalharmonics following [23].Any field on this background can be factorized into a part depending only from the timedirection, and another term living on the three-sphere. We focus on the latter factor.The three-sphere has isometry group G = SO (4) and local rotational invariance under H = SO (3) , hence it can be defined by the coset G/H = SO (4) /SO (3) . For convenience,we use the local isometry SO (4) ’ SU (2) × SU (2) to split the irreducible representationsof G into products of the irreducible representations of SU(2) , which are labelled by integerand half-integer spins J, ˜ J .
A basis for such a representation will be denoted by | J, m i| ˜ J , ˜ m i , with | m | ≤ J and | ˜ m | ≤ ˜ J .
For the local invariance, we denote the spin of the irreduciblerepresentation as L and the states as | Ln i , with the constraint | n | ≤ L. If we denote the generators of G with J i , ˜ J i and the generators of H with L i (in both cases i ∈ { , , } ), they are related by L i = J i + ˜ J i . In this way, we can introduce SU(2) Clebsch-Gordan coefficients to obtain the representations of H from the sum of the two representationsSU(2) composing G, yielding the expression | Ln ; J ˜ J i = X m, ˜ m C LnJm ; ˜ J ˜ m | J m i| ˜ J ˜ m i , (A.1)with the triangle inequalities | J − ˜ J | ≤ L ≤ J + ˜ J . (A.2)40pherical harmonics on S are defined starting from these basis and from the choice of arepresentative element of G/H.
While most of the results do not depend from the specificchoice of this representative, the rotation charges will. For this reason, we specify that weparametrize the unit three-sphere with coordinates d Ω = dψ + cos ψ dφ + sin ψ dφ . (A.3)In this way we find the group elementΥ(Ω) = e − iφ ( J − ˜ J ) e iφ ( J + ˜ J ) e − iψ ( J − ˜ J ) , (A.4)and the corresponding inverseΥ − (Ω) = e iψ ( J − ˜ J ) e iφ ( J − ˜ J ) e − iφ ( J + ˜ J ) . (A.5)The spherical harmonics are defined as Y LnJm ; ˜ J ˜ m (Ω) = s (2 J + 1)(2 ˜ J + 1)2 L + 1 h Ln ; J ˜ J | Υ − (Ω) | J m ; ˜ J ˜ m i , (A.6)with m, ˜ m being the eigenvalues of the generators J , ˜ J , respectively.At this point, we specify the previous decomposition for the fields of interest on the three-sphere: scalars, fermions and gauge fields. Since a scalar field is a singlet under the localrotations SO (3) , its spin is L = 0 and this immediately implies that J = ˜ J .
The decompositionis easily given byΦ a ( t, Ω) = X J,M Φ aJM ( t ) Y JM (Ω) , Φ † a ( t, Ω) = X JM Φ † JM,a ( t ) ¯ Y JM (Ω) , (A.7)where we denoted Y JM ≡ Y L =0 ,n =0 J,m ; J, ˜ m . (A.8)In the previous sums we collected the eigenvalues of the momenta as M = ( m, ˜ m ) , bothrunning from − J to J, while J itself runs over positive integers and half-integers.Spinor fields have L = , which allows for the possibilities to take momenta ( J + , J ) orviceversa, i.e. ( J, J + ) . In this case the mode expansion reads ψ Aα ( t, Ω) = X κ = ± X JM ψ AJM,κ ( t ) Y κJM,α (Ω) , ψ † ˙ α,A ( t, Ω) = X κ = ± X JM ψ † JM,κ,A ( t ) ¯ Y κJM, ˙ α (Ω) , (A.9)where we defined Y κ =1 JM,α ≡ Y L = ,αJ + ,m ; J, ˜ m , Y κ = − JM,α ≡ Y L = ,αJ,m ; J + , ˜ m . (A.10)While J runs again over positive integers and half-integers, now the momenta ( m, ˜ m ) aresummed from − U to U and − ˜ U to ˜ U , respectively, with U = J + κ and ˜ U = J + − κ . The gauge fields are vectors, and then they have L = 1 . This allows for even more possi-bilities, i.e. we can take ( J + 1 , J ) , ( J, J ) or (
J, J + 1) . Then their decomposition is A i ( t, Ω) = X ρ = − , , X JM A JM ( ρ ) ( t ) Y ρJM,i (Ω) , (A.11)41ith Y ρ =1 JM,i ≡ i Y L =1 ,iJ +1 ,m ; J, ˜ m , Y ρ =0 JM,i ≡ Y L =1 ,iJ,m ; J, ˜ m , Y ρ = − JM,i ≡ − i Y L =1 ,iJ,m ; J +1 , ˜ m . (A.12)In this case ( m, ˜ m ) run from − Q to Q and − ˜ Q to ˜ Q, respectively, with Q = J + ρ (1+ ρ )2 and˜ Q = J − ρ (1 − ρ )2 , and J is summed over positive integers and half-integers.Notice that while the components along the three-sphere of the gauge field A i are vectors,instead the temporal component A behaves as a scalar. Consequently, its decomposition onlyinvolves the harmonics with L = n = 0 , and reads A ( t, Ω) ≡ χ ( t, Ω) = X J,M χ JM ( t ) Y JM (Ω) . (A.13)In view of the manipulations with the Hamiltonian formalism, we also introduce the modeexpansion of the momenta, which we denote as Π ( F ) , being F ∈ { Φ , ψ, A } the field whose thespecific momentum is associated to. Since the orthonormality of the basis involves an innerproduct between spherical harmonics and their complex conjugate, it is convenient to choose a decomposition of spherical harmonics for the corresponding canonical momenta given byΠ a (Φ) ( t, Ω) = X J,M Π a (Φ) JM ( t ) ¯ Y JM (Ω) , Π † (Φ) a ( t, Ω) = X JM Π † (Φ) JM,a ( t ) Y JM (Ω) , (A.14)Π A ( ψ ) α ( t, Ω) = X κ = ± X JM Π A ( ψ ) JM,κ ( t ) ¯ Y κJM,α (Ω) , Π ( A ) i ( t, Ω) = X ρ = − , , X JM Π JM ( A )( ρ ) ( t ) ¯ Y ρJM,i (Ω) . (A.15)We also specify the mode expansion of the current and the Lagrange multiplier enteringEq. (2.8), where we apply the same convenient choice: j ( t, Ω) = X J,M j † JM ( t ) ¯ Y JM (Ω) , j i ( t, Ω) = X J,M,ρ j † JM ( ρ )) ( t ) ¯ Y ρJM,i (Ω) , (A.16) η ( t, Ω) = X J,M η † JM ( t ) ¯ Y JM (Ω) . (A.17) B Hamiltonian and conserved charges of N = 4 on S In view of the evaluation of BPS limits for the various sectors of N = 4 SYM, we collect theconventions about the relevant rotational and internal charges on the three-sphere S . Wemostly follow the same notation as [23].The free part of the action on R × S is given by S = Z R × S q − det g µν tr (cid:26) − ( ∇ µ Φ a ) † ∇ µ Φ a − Φ † a Φ a − iψ † A ¯ σ µ ∇ µ ψ A − F µν F µν (cid:27) , (B.1)where we remind that ∇ denotes the covariant derivative containing only the gravity contri-butions, without the minimal coupling with the gauge fields. We stress that excluding the In principle we can expand the fields on the three-sphere in terms of the spherical harmonics Y or in termsof their complex conjugate ¯ Y . The difference between them amounts to a phase and some change of the labels,as shown in (C.26). Therefore, the two choices correspond to a redefinition of the modes defining the expansionof the field. , Weyl fermions ψ and gauge fields A arerespectively given byΠ (Φ) a = 1 p − det g µν δSδ ˙Φ a = ˙Φ † a , Π a † (Φ) = 1 p − det g µν δSδ ˙Φ † a = ˙Φ a , Π ( ψ ) A = 1 p − det g µν δSδ ˙ ψ A = iψ † A , Π ( A )0 = 1 p − det g µν δSδ ˙ A = 0 , Π ( A ) i = 1 p − det g µν δSδ ˙ A i = F i . (B.2)From now on, we will avoid specifying the field F associated to the momentum Π ( F ) , sincethis will be clear from the context. Notice that the first order nature of the fermionic actionis responsible for obtaining a proportionality between the Weyl fermion and the hermitianconjugate of the corresponding momentum. This allows to choose if we want to express theHamiltonian and the charges of interest in terms of squares of the fields or of their momenta,or if we want mixed products of them.These momenta allow to compute the free Hamiltonian of the system by means of theLegendre transform, giving H = Z R × S q − det g µν tr (cid:26) | Π a | + |∇ i Π a | − iψ † A σ i ∇ i ψ A + Π i + 12 F ij (cid:27) . (B.3)The issues related to imposing the Coulomb gauge and the corresponding constraints, whichrequire to introduce the Dirac brackets, are discussed in section 2. Here we report the resultof such discussion: the ρ = 0 mode of the gauge field A i is vanishing, and the temporalcomponent A can be integrated out.In this way, after using the mode expansions (A.7), (A.9) and (A.12), we find the freeHamiltonian H = X J,m, ˜ m tr | Π Jm ˜ ma | + ω J | Φ Jm ˜ ma | − X κ = ± κω ψJ ψ † JM,κ,A ψ AJM,κ + X ρ = ± (cid:16) | Π Jm ˜ m ( ρ ) | + ω A,J | A Jm ˜ m ( ρ ) | (cid:17) , (B.4)where ω J ≡ J + 1 , ω ψJ ≡ J + 32 , ω A,J ≡ J + 2 . (B.5)A peculiarity of this free Hamiltonian is that while the scalar and gauge terms are manifestlypositive-definite, instead the fermionic part is apparently negative-definite when κ = 1 , i.e. we have H ( ψ )0 = X JM tr (cid:16) − ω ψJ ψ A † JM,κ =1 ψ AJM,κ =1 + ω ψJ ψ A † JM,κ = − ψ AJM,κ = − (cid:17) . (B.6)The reason for this apparent negativity of the fermionic term arises from the conventionsin [23], because after quantization it is required that the two chiralities of the fermions are43ecomposed as follows: ψ AJM,κ =1 = d A † J, − M e iω ψJ , ψ AJM,κ = − = b AJM e − iω ψJ . (B.7)Since one polarization acts as a creation operator and the other one as an annihilation oper-ator, in the end the Hamiltonian is positive definite.We find a manifestly positive definite expression even at the level of the classical action ifwe redefine ψ AJM,κ =1 → ψ A † J, − M,κ =1 , ψ A † JM,κ =1 → ψ AJ, − M,κ =1 . (B.8)It is important to notice that the redefinition also involves a change of sign for the orbitalmomentum eigenvalue M = ( m, ˜ m ) . This change can be easily understood if we think thatthe creation of a particle with momentum M is now interpreted as the annihilation of anantiparticle of momentum − M, ad viceversa.In this way, using the Grassmannian nature of the fermions, we find in terms of theredefined quantities that H ( ψ )0 = X JM tr (cid:16) − ω ψJ ψ AJ, − M,κ =1 ψ A † J, − M,κ =1 + ω ψJ ψ A † JM,κ = − ψ AJM,κ = − (cid:17) == X JM tr (cid:16) ω ψJ ψ A † JM,κ =1 ψ AJM,κ =1 + ω ψJ ψ A † JM,κ = − ψ AJM,κ = − (cid:17) = X κ = ± X JM tr (cid:16) ω ψJ ψ † JM,κ,A ψ AJM,κ (cid:17) . (B.9)In the first step we sent the index M → − M due to the symmetry of the range of summation.We observe that the sign given by the factor − κ in the free Hamiltonian disappears, and themap from the previous notation to the new conventions implies X JM X κ = ± ψ † JM,κ,A ψ AJM,κ → X JM X κ = ± − κψ † JM,κ,A ψ AJM,κ . (B.10)From now on, all the quantities involving fermionic fields will be computed after applyingthe prescription (B.8). We will add some additional comments on these terms only whencomputing the Cartan charges associated to rotation and R-symmetry.Now we compute the relevant currents corresponding to the symmetries of the action(B.1). We start with the canonical energy-momentum tensor T µν ≡ T (Φ) µν + T ( ψ ) µν + T ( A ) µν + g µν √− g L , (B.11)where L is the Lagrangian density and T (Φ) µν = ( ∂ µ Φ a ) † ∂ ν Φ a + ( ∂ µ Φ a ) † ∂ ν Φ a , (B.12) T ( ψ ) µν = − i ψ † A ¯ σ µ ( ∇ ν ψ A ) + i ( ∇ ν ψ A ) † ¯ σ ν ψ A , (B.13) T ( A ) µν = F σµ F νσ . (B.14)The conserved charges corresponding to the commuting rotation generators S , S on thethree-sphere are defined as S i = Z S d Ω T i , (B.15)being d Ω the volume form on the three-sphere. Introducing the decomposition of the fieldsinto spherical harmonics from Appendix A, we get S i ≡ S (Φ) i + S ( ψ ) i + S ( A ) i , (B.16)44here S (Φ)1 = P J,M i ( ˜ m − m ) tr (cid:16) Φ JM Π JM − Φ † JM Π † JMφ (cid:17) , (B.17) S ( ψ )1 = P JM P κ = ± ( ˜ m − m ) tr (cid:16) ψ † JM,κ,A ψ AJM,κ (cid:17) , (B.18) S ( A )1 = P J,m, ˜ m P ρ = − , i ( ˜ m − m ) tr (cid:16) A Jm ˜ m ( ρ ) Π Jm ˜ m ( ρ ) − A † Jm ˜ m ( ρ ) Π † Jm ˜ m ( ρ ) (cid:17) , (B.19)and S (Φ)2 = P J,M i ( m + ˜ m ) tr (cid:16) Φ JM Π JM − Φ † JM Π † JMφ (cid:17) , (B.20) S ( ψ )2 = P JM P κ = ± ( m + ˜ m ) tr (cid:16) ψ † JM,κ,A ψ AJM,κ (cid:17) , (B.21) S ( A )2 = P J,m, ˜ m P ρ = − , i ( m + ˜ m ) tr (cid:16) A Jm ˜ m ( ρ ) Π Jm ˜ m ( ρ ) − A † Jm ˜ m ( ρ ) Π † Jm ˜ m ( ρ ) (cid:17) . (B.22)In order to derive the previous action of the derivatives on the spherical harmonics, it is crucialto use the specific group elemen on the three-sphere G/H = SO (4) /SO (3) in Eq. (A.4), sincederivatives along the angular directions are required.Notice that the expression for the fermionic charge is exactly the same before and afterthe redefinition (B.8), as can be seen by direct evaluation. For this equivalence to hold it iscrucial to use the flipping of M. The other relevant charges are associated to the Cartan subalgebra of the global R-symmetry of the action. They can be written as Q a = Q (Φ) a + Q ( ψ ) a , (B.23)where Q (Φ) a = i P J,M tr (cid:16) Φ JMa Π JMa − Φ † JMa Π † ,JMa (cid:17) , (B.24) Q ( ψ ) a = P κ = ± P JM κ tr (cid:16) ψ † JM,κ,A ( T a ) AB ψ BJM,κ (cid:17) . (B.25)The matrices of the Cartan subalgebra in the fundamental representation of SU(4) are T ≡
12 diag { , − , − , } , T ≡
12 diag { , − , , − } , T ≡
12 diag { , , − , − } . (B.26)Notice that in the end all the explicit κ dependence in the Hamiltonian and the other conservedquantities is only isolated to the R-charges. Weights
In order to sistematically explore the near-BPS limits of N = 4 SYM on R × S , it is convenientto list the set of letters of the theory, which is composed by 6 complex scalars, 16 complexGrassmannian fields and 6 independent gauge field strength components, plus the descendantsobtained by acting with the 4 components of the covariant derivatives. We assign weightsunder the subgroups SO(4) (rotations) and SU(4) (R-symmetry) of PSU(2 , | , following theconventions of reference [24]. We start with the weights under SU(4) , which are reported inTable 2, 3 and 4. The field strength and the covariant derivatives d , d , ¯ d , ¯ d are unchargedunder this symmetry. 45 X W ¯ Z ¯ X ¯ W (1 , ,
0) (0 , ,
0) (0 , ,
1) ( − , ,
0) (0 , − ,
0) (0 , , − χ , χ χ , χ χ , χ χ , χ (cid:16) , , (cid:17) (cid:16) , − , − (cid:17) (cid:16) − , , − (cid:17) (cid:16) − , − , (cid:17) Table 3: Fermionic SU(4) weights for fermions χ in the notation of [24]In order to make contact with the notation used in this work, we read off the SU(4) R-symmetry weights of all fields using (B.24), (B.25) and (B.26). We list them in tables 5 and6, respectively. By looking at the dynamical modes that we describe in the sectors of section2, we verify that the results are consistent with the list of surviving field in the limits givenin reference [24], see subsection below.It is also convenient to compare this notation with the conventions of the paper [23],where the antisymmetric representation is instead used for the scalar fields. In Eq. (2.19) ofreference [23] the transformation properties of the antisymmetric tensor and of the fermionsunder R-symmetry are reported: δ R X AB = iT AC X CB + T BC X AC , δ R ψ A = iT AB ψ B . (B.27)The weights of scalars can be immediately deduced from this rule and from the basis for thegenerators (B.26). The transformation of the fermionic fields is the same used here to derivethe expression (B.25), and then they agree. The comparison is consistent if we chooseΦ = X , Φ = X † , Φ = X † . (B.28)This is the dictionary that we will use throughout all the computations in the present work.In addition, the fields also carry charge under SO (4) rotations, except for the scalars. Welist their quantum number in the subsection below, referring to the specific sectors where wetake the limits. Charges for the specific near-BPS limits
From Eq. (B.16) applied to the specific cases, we identify S = − m + ˜ m , S = m + ˜ m (B.29)¯ χ , ¯ χ ¯ χ , ¯ χ ¯ χ , ¯ χ ¯ χ , ¯ χ (cid:16) − , − , − (cid:17) (cid:16) − , , (cid:17) (cid:16) , − , (cid:17) (cid:16) , , − (cid:17) Table 4: Fermionic SU(4) weights for fermions ¯ χ in the notation of [24]46 Φ Φ (1 , ,
0) (0 , ,
0) (0 , , ψ ψ ψ ψ κ = 1 (cid:16) , , (cid:17) (cid:16) − , − , (cid:17) (cid:16) − , , − (cid:17) (cid:16) , − , − (cid:17) κ = − (cid:16) − , − , − (cid:17) (cid:16) , , − (cid:17) (cid:16) , − , (cid:17) (cid:16) − , , (cid:17) Table 6: Fermionic SU(4) weightsHere we list the rotation and the R-symmetry charges for all the limits considered in thiswork.• In the bosonic SU(1 ,
1) sector we have a surviving dynamical scalar with derivatives d n Z. The associated momenta and charges are − m = ˜ m = J , S = 2 J , S = 0 , (B.30)( Q , Q , Q ) = (1 , , , (B.31)with 2 J ∈ N and n = 2 J .• In the fermionic SU(1 ,
1) limit we have the fermion with derivatives d n χ with quantumnumbers κ = 1 , m = − J − , ˜ m = − J , S = 2 J + , S = − , (B.32)( Q , Q , Q ) = (cid:16) , , (cid:17) , (B.33)with 2 J ∈ N and n = 2 J .• The SU(1 , |
1) sector simply contains the union of the degrees of freedom in the SU(1 , , |
2) sector, in addition to the abovementioned fields d n Z, d n χ , there areone more scalar field with derivatives d n X and one more fermion with derivatives d n ¯ χ . The additional scalar has the same quantum numbers as d n Z, i.e. − m = ˜ m = J , S = 2 J , S = 0 , (B.34)( Q , Q , Q ) = (1 , , , (B.35)with 2 J ∈ N and n = 2 J .The fermion d n ¯ χ has instead different momenta and charges, given by κ = − , m = − J , ˜ m = J + , S = 2 J + , S = − , (B.36)( Q , Q , Q ) = (cid:16) , , − (cid:17) , (B.37)with 2 J ∈ N and n = 2 J . 47 nteracting Hamiltonian Using the decomposition into spherical harmonics on the three-sphere, we derive the interact-ing Hamiltonian of N = 4 SYM on R × S . The entire expression can be found in [23], butwe report here the result using our notation. Due to the redefinition of fermions (B.8) andthe dictionary (B.28), it is convenient to introduce the notations( Z a ) JM ≡ (Φ ) JM ( − m − ˜ m (Φ † ) J, − M ( − m − ˜ m (Φ † ) J, − M , (B.38)(Ψ A ) J,M,κ =1 ≡ ( ψ † A ) J, − M,κ =1 , (Ψ A ) J,M,κ = − ≡ ( ψ A ) J,M,κ = − . (B.39)Notice that these definitions account precisely for the different interpretation of scalars andfermions with respect to reference [23], see i.e. the action of complex conjugation on sphericalharmonics described in Eq. (C.26).The result is: H int = X J i ,M i ,κ i ,ρ i tr n ig C J M J M ; JM χ JM (cid:16) [( Z † a ) J M , (Π (Φ) † a ) J M ] + [ Z aJ M , Π a (Φ) J M ] (cid:17) − g q J ( J + 1) D J M J M JMρ A JM ( ρ ) [ Z aJ M , ( Z † a ) J M ]+ g F J M κ J M κ ; JM χ JM { (Ψ † A ) J M κ , Ψ AJ M κ } + g G J M κ J M κ ; JMρ A JM ( ρ ) { (Ψ † A ) J M κ , Ψ AJ M κ } + g C J M J M ; JM C J M J M ; JM [ Z aJ M , ( Z † a ) J M ][ Z bJ M , ( Z † b ) J M ] −√ ig ( − − m + ˜ m + κ F J , − M ,κ J M κ ; JM ψ J M κ [( Z a ) JM , Ψ aJ M κ ]+ √ ig ( − − m + ˜ m + κ F J , − M ,κ J M κ ; JM (cid:15) abc Ψ aJ M κ [( Z † b ) JM , Ψ cJ M κ ]+ √ ig ( − m − ˜ m + κ F J M κ J , − M ,κ ; JM (Ψ † ) J M κ [( Z † a ) JM , (Ψ † a ) J M κ ] −√ ig ( − m − ˜ m + κ F J M κ J , − M ,κ ; JM (cid:15) abc (Ψ † a ) J M κ [( Z b ) JM , (Ψ † c ) J M κ ]+ ig D JMJ M ρ ; J M ρ χ JM [Π J M ( ρ ) , A J M ( ρ ) ]+ g C JMJ M ; J , − M D JM ; J M ρ ; J M ρ [ A J M ( ρ ) , Z aJ M ][ A J M ( ρ ) , ( Z † a ) J M ]+2 igρ ( J + 1) E J M ρ ; J M ρ ; J M ρ A J M ( ρ ) [ A J M ( ρ ) , A J M ( ρ ) ] − g D JMJ M ρ ; J M ρ D JM ; J M ρ ; J M ρ [ A J M ( ρ ) , A J M ( ρ ) ][ A J M ( ρ ) , A J M ( ρ ) ] − g q J ( J + 1) D J M ; J M JMρ χ J M [ χ J M , A JM ( ρ ) ]+ g C JMJ M ; J M D JM ; J M ρ ; J M ρ [ χ J M , A J M ( ρ ) ][ χ J M , A J M ( ρ ) ]+ g C JMJ M ; J M C JM ; J M ; J M [ χ J M , Z aJ M ][ χ J M , ( Z † a ) J M ] o . (B.40)The notation used is the following. The initial sum represents a summation over all contractedindices: momenta ( J, M ) , labels for the spherical harmonics involving fermions κ and gaugefields ρ, and indices of the fields under SU(4) R-symmetry. The Yukawa term contains sumsof the spinors over only the subset a ∈ { , , } and the Levi-Civita symbol (cid:15) abc is defined in48uch a way that (cid:15) = 1 . In order to avoid confusion, we specified that Π Φ a are the canonicalmomenta associated to the scalar fields Φ a , while Π ( ρ ) is the symplectic partner of the gaugefield A ( ρ ) . The terms involving the gauge fields, except for the terms contributing to the bosonic andfermionic currents, are not needed for the near-BPS limits included in this work, but are putfor completeness. In order to derive from this expression the relevant contributions to theinteracting Hamiltonians in section 2, few simplifications still need to be performed, in orderto obtain the fields Φ a , ψ A from the variables (B.38) and (B.39). The simplified expressionsare written explicitly in section 2 for each case considered. C Properties of spherical harmonics and Clebsch-Gordan co-efficients
Definition of the Clebsch-Gordan coefficients
We give the explicit definitions of the Clebsch-Gordan coefficients which are used to computethe interacting Hamiltonians in section 2. They were previously given e.g. in [23]. C J M J M ; JM = s (2 J + 1)(2 J + 1)2 J + 1 C J m J m ; Jm C J ˜ m J ˜ m ; J ˜ m , (C.1) D J M J M ρ ; JMρ = ( − ρ ρ +1 q J + 1)(2 J + 2 ρ + 1)(2 J + 1)(2 J + 2 ρ + 1) × C J ,m Q ,m ; Q,m C J , ˜ m ˜ Q , ˜ m ; ˜ Q, ˜ m Q ˜ Q Q ˜ Q J J , (C.2) E J M ρ ; J M ρ ; JMρ = q J + 1)(2 J + 2 ρ + 1)(2 J + 1)(2 J + 2 ρ + 1)(2 J + 1)(2 J + 2 ρ + 1) × ( − − ρ ρ ρ +12 Q ˜ Q Q ˜ Q Q ˜ Q Q Q Qm m m ! ˜ Q ˜ Q ˜ Q ˜ m ˜ m ˜ m ! , (C.3) F J M κ J M κ ; JM =( − ˜ U + U + J + q (2 J + 1)(2 J + 1)(2 J + 2) × C U ,m U ,m ; J,m C ˜ U , ˜ m ˜ U , ˜ m ; J, ˜ m ( U ˜ U ˜ U U J ) , (C.4) G J M κ J M κ ; JMρ =( − ρ q J + 1)(2 J + 2)(2 J + 1)(2 J + 2 ρ + 1) × C U ,m U ,m ; Q,m C ˜ U , ˜ m ˜ U , ˜ m ; ˜ Q, ˜ m U ˜ U U ˜ U Q ˜ Q , (C.5)where we defined the quantities U ≡ J + κ + 14 , ˜ U ≡ J + 1 − κ , Q ≡ J + ρ ( ρ + 1)2 , ˜ Q ≡ J + ρ ( ρ − . (C.6)49roperties of 9-j and 6-j Wigner symbols were used to write the coefficient F in this form,but the expression is still completely general.In view of the crossing relations that we will derive, it is also useful to record the integralrepresentation of the previous Clebsch-Gordan coefficients as products of spherical harmonicson the three-sphere. Precisely, they are given by C J M J M ; JM = Z S d Ω ¯ Y J M Y J M Y JM , (C.7) D J M J M ρ ; JMρ = Z S d Ω ¯ Y J M Y ρ J M i Y ρJMi , (C.8) E J M ρ ; J M ρ ; JMρ = Z S d Ω (cid:15) ijk Y ρ J M i Y ρ J M j Y ρJMk , (C.9) F J M κ J M κ ; JM = Z S d Ω ¯ Y κ J M α Y κ J M α Y JM , (C.10) G J M κ J M κ ; JMρ = Z S d Ω σ iαβ ¯ Y κ J M α Y κ J M β Y ρJMi , (C.11)where ¯ Y denotes the complex conjugate of the harmonics Y . We reported here the coefficient E for completeness, but it will never enter in any interacting Hamiltonian for the near-BPSlimits considered in this work because it only couple terms containing dynamical gauge fields,while in the SU(1 ,
1) sector and its generalizations the gauge field always decouples.At this point we start specializing these definitions to the cases of interest for the near-BPS limits. The crossing relations between them will allow to analitically solve the sums overintermediate momenta J appearing in the computation of the interacting Hamiltonian.We start from C , which enters all the computations of the various sectors only via theprescription of momenta (2.36). Using the definition (C.1) and specializing to this case, weeasily obtain by direct computation C J J ; JM ≡ C J , − J ,J J , − J ,J ; Jm ˜ m = ( − J − J + J s (2 J + 1)(2 J + 1)2 J + 1 (2 J + 1)!(2 J )!( J + J − J )!( J + J + J + 1)! . (C.12) Crossing relations at saturated angular momenta – C and D We consider the definition (C.2) and we specialize the momenta to the assignments in Eq. (2.36)with ρ = ± . In fact, these are the only two cases of interest for the computation of the in-teracting part of the Hamiltonian mediated by the non-dynamical gauge field. The explicitexpressions are D J J ; Jm ˜ m,ρ =1 = − i ( − J − J + J s (2 J + 1)( J + ∆ J + 1)( J − ∆ J + 1) J ( J + 1)( J + 1)(2 J + 1) × (2 J )!(2 J + 1)!2( J + 1 + J + J )!( J + J − J − , (C.13)where ∆ J = J − J , and D J J ; Jm ˜ m,ρ = − = −D J J ; Jm ˜ m,ρ =1 . (C.14)50t is convenient for the following manipulations to factorize from this formula appropriatefactors of the Clebsch-Gordan coefficient C computed above. We find D J J ; Jm ˜ m,ρ =1 = i J − J − J ) s ( J + ∆ J + 1)( J − ∆ J + 1) J ( J + 1)(2 J + 1)( J + 1) C J J ; Jm ˜ m == i J + J + J + 2) s ( J + ∆ J + 1)( J − ∆ J + 1) J ( J + 1)(2 J + 3)( J + 1) C J J ; J +1 ,m ˜ m . (C.15)At this point, we consider appropriate quadratic combinations of the Clebsch-Gordan coeffi-cients C , D in view of finding simplifications which allow to solve the sum over intermediatemomenta J. We define the quantities A J , J J , J ; Jm ˜ m = (cid:18) ω J + ω J )( ω J + ω J )4 J ( J + 1) (cid:19) C J J ; Jm ˜ m C J J ; Jm ˜ m (C.16)and B J , J J , J ; Jm ˜ mρ = 16 ω A,J − ( m − ˜ m ) q J ( J + 1) J ( J + 1) D J J ; Jm ˜ mρ ¯ D J J ; Jm ˜ mρ , (C.17)which for ρ = ± B J , J J , J ; Jm ˜ m,ρ =1 = (2 + J + J + J )(2 + J + J + J )( J + 1)(2 J + 3) C J J ,J +1 m ˜ m C J J ,J +1 m ˜ m , (C.18) B J , J J , J ; Jm ˜ m,ρ = − = ( J + J − J )( J + J − J )( J + 1)(2 J + 1) C J J ,Jm ˜ m C J J ,Jm ˜ m . (C.19)Simple algebraic manipulations now give rise to the relation B J , J J , J ; Jm ˜ m,ρ = − + B J , J J , J ; J − m ˜ m,ρ =1 = A J , J J , J ; Jm ˜ m , (C.20)valid for J ≥ X Jm ˜ m (cid:18) ω J + ω J )( ω J + ω J )4 J ( J + 1) (cid:19) C J J ,Jm ˜ m C J J ,Jm ˜ m − X ρ = ± ω A,J − ( m − ˜ m ) q J ( J + 1) J ( J + 1) D J J ; Jm ˜ mρ ¯ D J J ; Jm ˜ mρ ! = X J ≥ J min ( ρ ) (cid:16) A J , J J , J ; J, − ∆ J, ∆ J − B J , J J , J ; J, − ∆ J, ∆ J,ρ = − − B J , J J , J ; J, − ∆ J, ∆ J,ρ =1 (cid:17) , (C.21)where here ∆ J = J − J = J − J . In general, if we also define ∆ m = m − m = m − m (and similarly for the ˜ m ), we should be careful in distinguishing the cases | ∆ J | < | ∆ m | and | ∆ J | ≥ | ∆ m | , because the triangle inequalities and the constraints on the eigenvalues ofmomenta imply that the lower bound of summation for J changes.In this case, however, all the momenta are fixed and we notice that ∆ m = − ∆ ˜ m = − ∆ J, so there is only one case to consider. The eqs.(C.16) - (C.18) imply that J min = | ∆ J | for allthe terms in (C.21). Shifting J → J − J > | ∆ J | ,leaving the final expression X J ≥| ∆ J | (cid:16) A J , J J , J ; J, − ∆ J, ∆ J − B J , J J , J ; J, − ∆ J, ∆ J,ρ = − − B J , J J , J ; J, − ∆ J, ∆ J,ρ =1 (cid:17) = B J , J J , J ; | ∆ J |− , − ∆ J, ∆ J,ρ =1 , (C.22)51here we have used Eq. (C.20) to simplify the result.Explicit evaluation yields B J , J J , J ; | ∆ J |− , − ∆ J, ∆ J,ρ =1 = (1 + | ∆ J | + J + J )(1 + | ∆ J | + J + J ) | ∆ J | (2 | ∆ J | + 1) C J J , | ∆ J | , − ∆ J, ∆ J C J J , | ∆ J | , − ∆ J, ∆ J . (C.23) Crossing relations at saturated angular momenta – C and F We start by deriving a couple of properties which relate Clebsch-Gordan coefficients F withdifferent assignments of momenta, useful to obtain the simplification in Eq. (2.63): F J ,M ,κ J , − M ,κ ; JM = ( − m − ˜ m F J , − M ,κ J ,M ,κ ; J, − M , (C.24) F J ,M ,κ J ,M ,κ ; JM = ( − m − ˜ m + m − ˜ m + κ κ F J , − M ,κ J , − M ,κ ; JM . (C.25)These identities can be easily derived by using the integral representation (C.10) combinedwith the properties of spherical harmonics under complex conjugation, i.e. ¯ Y JM = ( − m − ˜ m Y J, − M , ¯ Y κJMα = ( − m − ˜ m + κα +1 Y κJ, − M, − α . (C.26)Now we focalize instead to the specific cases of interest for the computation of the Hamiltonianin the near-BPS bounds of interest. Contrarily to the bosonic case, the Clebsch-Gordancoefficients involving spherical harmonics of fermions appear in the sectors with two differentpossibilities, see the analysis of the PSU(1 , |
2) sector: with chirality κ = 1 and momenta( m, ˜ m ) = ( − J − , J ) or with κ = − − J, J + ) . Due to the redefinition ofthe former in Eq. (B.8), we get the two quantities F J ,J + , − J ,κ =1 J ,J + , − J ,κ =1; Jm ˜ m , F J , − J ,J + ,κ = − J , − J ,J + ,κ = − Jm ˜ m . (C.27)Looking at the definition (C.4), we notice that the two expressions are related. In fact, thetwo factors of SU(2) Clebsch-Gordan coefficients C entering the definition of F are simplyexchanged in the two cases, while the Wigner 6-j symbols are the same due to the symmetryunder the interchange of two elements of a line with the other one ( J + J J J + J ) = ( J J +
12 12 J + J J ) . (C.28)Finally, the triangle inequalities and the conditions on integer sums of momenta coincide inthe two cases, thus we conclude that F J ,J + , − J ,κ =1 J ,J + , − J ,κ =1; Jm ˜ m = F J , − J ,J + ,κ = − J , − J ,J + ,κ = − Jm ˜ m . (C.29)More generally, the interactions of the PSU(1 , |
2) sector also involve terms where the chirali-ties κ , κ entering the definition of F assume both values ± . In such case, it is convenient tofind additional crossing relations between the coefficients with various assignments of momentaand chiralities. 52e follow the same steps depicted above, but now we notice that when κ , κ are not fixedthe triangle conditions on momenta imply J + J + J ∈ Z , while when κ = − κ we have J + J + J + ∈ Z . This gives only two possibilities for the overall sign, which are F J m ˜ m κ J m ˜ m κ ; Jm ˜ m = F J , ˜ m ,m , − κ J , ˜ m ,m , − κ ; J ˜ mm if κ = κ −F J , ˜ m ,m , − κ J , ˜ m ,m , − κ ; J ˜ mm if κ = − κ . (C.30)In fact, the relation (C.29) corresponds to the first case with κ = κ = 1 . We observe that under the exchange m ↔ ˜ m, κ → − κ, the following identity holds: κω ψJ − ( m − ˜ m ) → − (cid:16) κω ψJ − ( m − ˜ m ) (cid:17) . (C.31)This expression appears at the denominator of a relevant interaction in the PSU(1 , |
2) sector.We thus study in details the following expression when κ = κ X κ = ± F J , − m , − ˜ m ,κ Jm ˜ mκ ; J m ˜ m F J , − m , − ˜ m ,κ Jm ˜ mκ ; J m ˜ m κω ψJ − ( m − ˜ m ) = − X κ = ± F J , − ˜ m , − m , − κ J ˜ mmκ ; J ˜ m m F J , − ˜ m , − m , − κ J ˜ mmκ ; J ˜ m m κω ψJ − ( m − ˜ m ) , (C.32)and when κ = − κ X κ = ± F J , − m , − ˜ m ,κ Jm ˜ mκ ; J m ˜ m F J , − m , − ˜ m ,κ Jm ˜ mκ ; J m ˜ m κω ψJ − ( m − ˜ m ) = X κ = ± F J , − ˜ m , − m , − κ J ˜ mmκ ; J ˜ m m F J , − ˜ m , − m , − κ J ˜ mmκ ; J ˜ m m κω ψJ − ( m − ˜ m ) . (C.33)Summarizing, we found that there is only one independent assignment of momenta whichis relevant to derive the interacting Hamiltonians of section 2, identified by the short-handnotation (2.61). We use the definition (C.4) to find F ¯ J ¯ J ; Jm ˜ m = C J J ; Jm ˜ m . (C.34)Surprisingly, when momenta are saturated in this way, we find an equivalence between theClebsch-Gordan coefficients involving only scalar harmonics, and this one involving mixedproducts between scalar and spinorial harmonics. Crossing relations at saturated angular momenta – C and G The Clebsch-Gordan coefficient G only appears in the computation of terms involving thefermionic current. This restricts the assignments on momenta of interest to the two cases G J ,J + , − J ,κ =1 J ,J + , − J ,κ =1; Jm ˜ mρ , G J , − J ,J + ,κ = − J , − J ,J + ,κ = − Jm ˜ mρ . (C.35)The strategy is to consider the general definition (C.5) and apply the symmetry properties ofSU(2) Clebsch-Gordan and Wigner 9-j symbols to find a relation between the two possibilities.Then, putting the assignments of momenta, we will relate the specific cases with ρ = ± C . For convenience, we write here the two explicit expressions: G J ,J + , − J ,κ =1 J ,J + , − J ,κ =1; JMρ = ( − ρ q J + 1)(2 J + 2)(2 J + 1)(2 J + 3) × C J + ,J + J + ,J + ; Q,J − J C J , − J J , − J ; ˜ Q,J − J J + J J + J Q ˜ Q , (C.36)53nd the other one G J , − J ,J + ,κ = − J , − J ,J + ,κ = − Jm ˜ mρ = ( − ρ q J + 1)(2 J + 2)(2 J + 1)(2 J + 3) × C J , − J J , − J ; Q,J − J C J + ,J + J + ,J + ; ˜ Q,J − J J J +
12 12 J J +
12 12 Q ˜ Q , (C.37)where in both cases we used the labels (C.6).Interestingly, the two expressions are almost the same: the prefactors coincide and the 9-jsymbol satisfies the property J + J J + J Q ˜ Q = ( − J + J + J ) J J +
12 12 J J +
12 12 Q ˜ Q . (C.38)Since for all the admitted choices of ρ we have J + J + J ∈ Z , the prefactor is 1 and thetwo expressions coincide. Thus the only difference between the two cases is due to the SU(2)Clebsch-Gordan coefficients, which however are simply exchanged if we also send Q ↔ ˜ Q. Since the interacting Hamiltonian contains only these quantities with ρ = ± , we havethat Q, ˜ Q assume in the two cases the values J + 1 , J. This implies the simple relation G J ,J + , − J ,κ =1 J ,J + , − J ,κ =1; Jm ˜ mρ = −G J , − J ,J + ,κ = − J , − J ,J + ,κ = − Jm ˜ m − ρ . (C.39)The different sign arises due to the factor ( − ρ , which gives an opposite sign to the imaginaryunit when considering ρ = ± . Then we can simply focus on one specific choice of the momenta and compute explicitlythe coefficient G , e.g. in the case in Eq. (2.61). We find G ¯ J ¯ J ; Jm ˜ m,ρ =1 = i s ( J + ∆ J + 1)( J − ∆ J + 1)( J + 1)(2 J + 1) C J J ; Jm , (C.40) G ¯ J ¯ J ; Jm ˜ m,ρ = − = − i s ( J + ∆ J + 1)( J − ∆ J + 1)( J + 1)(2 J + 3) C J J ; J +1 ,m . (C.41) Crossing relations at saturated angular momenta – F and G In this section we put together the identities between F , G with C in order to obtain a sim-plification for the terms involving fermions mediated by the non-dynamical gauge fields. Theinteraction of interest is X J i ,J,m, ˜ m J ( J + 1) F ¯ J ¯ J ,Jm ˜ m F ¯ J ¯ J ,Jm ˜ m − X ρ = ± ω A,J − ( m − ˜ m ) ) G ¯ J ¯ J ; Jm ˜ mρ ¯ G ¯ J ¯ J ; Jm ˜ mρ . (C.42)Using the relations (C.34), (C.40), (C.41) and splitting the sum over ρ = ± X J,m, ˜ m (cid:18) J ( J + 1) C J J ; Jm ˜ m C J J ; Jm ˜ m − J + 1)(2 J + 3) C J J ; J +1 ,m, ˜ m C J J ; J +1 ,m, ˜ m − J + 1)(2 J + 1) C J J ; Jm ˜ m C J J ; Jm ˜ m (cid:19) . (C.43)54ow we define the convenient quantities P J ; J J ; J ; Jm ˜ m = J ( J +1) C J J ; Jm ˜ m C J J ; Jm ˜ m , (C.44) Q J ; J J ; J ; Jm ˜ m,ρ = − = − J +1)(2 J +3) C J J ; J +1 ,m, ˜ m C J J ; J +1 ,m, ˜ m , (C.45) Q J ; J J ; J ; Jm ˜ m,ρ =1 = − J +1)(2 J +1) C J J ; Jm ˜ m C J J ; Jm ˜ m . (C.46)It can be shown that P J ; J J ; J ; Jm ˜ m + Q J ; J J ; J ; J − ,m, ˜ m,ρ = − + Q J ; J J ; J ; Jm ˜ m,ρ =1 = 0 . (C.47)The sum (over an appropriate interval) of the previous quantities exactly gives the termmediated by the non-dynamical gauge field for the SU(1 ,
1) fermionic sector, see Eq. (2.66): X J ≥ J min ( ρ ) (cid:16) P J ; J J ; J ; J, − ∆ J, ∆ J + Q J ; J J ; J ; J, − ∆ J, ∆ J,ρ = − + Q J ; J J ; J ; J, − ∆ J, ∆ J,ρ =1 (cid:17) , (C.48)where we defined∆ J = J − J = J − J , ∆ m = m − m = m − m . (C.49)The lower extremum of summation plays a crucial role for the simplifications below. As inthe previous cases considered in this Appendix, the assignments of momenta completely fix∆ m = − ∆ ˜ m = − ∆ J, which implies that we need to consider only one possibility for theendpoints of summation.Indeed, all the sums start from the same value J min = | ∆ J | , and the shift J → J − Q ρ =1 changes the lower endpoint of its summation to J min = | ∆ J | − . In this way,using Eq. (C.47), we get a remarkable simplification which only leaves a non-vanishing termcoming from the boundary of summation X J ≥| ∆ J | (cid:16) P J ; J J ; J ; J, − ∆ J, ∆ J + Q J ; J J ; J ; J, − ∆ J, ∆ J,ρ = − + Q J ; J J ; J ; J, − ∆ J, ∆ J,ρ =1 (cid:17) == −Q J ; J J ; J ;∆ J − , − ∆ J, ∆ J,ρ = − . (C.50)In particular, we can explicitly evaluate this last term to obtain an expression in terms of thecoefficient C , i.e. we obtain −Q J ; J J ; J ;∆ J − , − ∆ J, ∆ J,ρ = − = 18 | ∆ J | (2 | ∆ J | + 1) C J J ; | ∆ J | , − ∆ J, ∆ J C J J ; | ∆ J | , − ∆ J, ∆ J . (C.51)Due to Eq. (C.39), it is clear that the same procedure can be applied to the case with adynamical fermion having κ = − . The difference in such case is that the shift J → J − ρ = 1 . The symmetryof the problem guarantees that the final result is the same, as it is explained for the spherereduction in the PSU(1 , |
2) sector.
Crossing relations at saturated angular momenta – products of C , D , F and G In this subsection we show another remarkable simplification for the sum over J of the mixedbosonic-fermionic term mediated by the non-dynamical gauge field.55e refer to Eq. (2.70), where however due to symmetry reasons it is sufficient to consideronly half of the terms. We thus define S J , J J , J ; JM ≡ J + J +14 J ( J +1) C J J ; JM F ¯ J ¯ J ; JM (C.52) T J , J J , J ; JMρ ≡ √ J ( J +1) ω A,J − ( m − ˜ m ) ( ¯ D J J ; JMρ G ¯ J ¯ J ,JMρ + D J J ,JMρ ¯ G ¯ J ¯ J ; JMρ ) . (C.53)We can write these combinations only in terms of the Clebsch-Gordan coefficient C by meansof the crossing relations proved in this Appendix. The result is S J , J J , J ; JM = J + J +14 J ( J +1) C J J ; JM C J J ; JM , (C.54) T J , J J , J ; JMρ = − = − J + J + J +24( J +1)(2 J +3) C J J ; J +1 ,M C J J ; J +1 ,M , (C.55) T J , J J , J ; JMρ =1 = − J + J − J J +1)(2 J +1) C J J ; JM C J J ; JM . (C.56)As we learnt from previous example, the strategy is to send J → J − ρ = − S J , J J , J ; JM + T J , J J , J ; J − ,Mρ = − + T J , J J , J ; JMρ = − = 0 . (C.57)In this way, we find that the sum over J required in Eq. (2.70) reduces to a boundary term.Indeed, we obtain X J ≥| ∆ J | (cid:16) S J ; J J ; J ; J, − ∆ J, ∆ J + T J ; J J ; J ; J, − ∆ J,ρ = − + T J ; J J ; J m ; J, − ∆ J,ρ =1 (cid:17) == −T J ; J J ; J ;∆ J − , − ∆ J,ρ = − = J + J + ∆ J | ∆ J | (2 | ∆ J | + 1) C J J ; | ∆ J | C J J ; | ∆ J | . (C.58)The same result also applies to the analog term involving fermions with κ = − , |
2) sector, as can be seen by applying (C.14), (C.29) and (C.39). The onlydifference for the fermions with opposite chirality is that we need instead to shift the termswith ρ = 1 , but the procedre is formally the same, as well as the final result that we obtain. D Algebra and oscillator representation
The oscillator representation [12] is a convenient way to represent the set of letters of N = 4SYM and its superconformal algebra u (2 , | . In this Appendix we review such a representa-tion, following the conventions of [24].We consider two sets of bosonic oscillators a α , b ˙ α with four dimensional spinorial indices α, ˙ α ∈ { , } and one fermionic oscillator c a with a ∈ { , , , } satisfying the canonicalcommutation relations h a α , a † β i = δ αβ , h b ˙ α , b † ˙ β i = δ ˙ α ˙ β , { c a , c † b } = δ ab . (D.1)We conveniently introduce the following notation to denote the number operators: a α ≡ a † α a α , b ˙ α ≡ b † ˙ α b ˙ α , c a ≡ c † a c a , (D.2)with no sum over the indices. 56hese oscillators can be combined in order to define the generators of the algebra and thephysical states. We have the 6 generators of the so (4) subalgebra L αβ = a † β a α − a + a δ αβ , ˙ L ˙ α ˙ β = b † ˙ β b ˙ α − b + b δ ˙ α ˙ β , (D.3)and 15 generators for the su (4) subalgebra R ab = c † b c a − δ ab X d =1 c d . (D.4)For the purposes of this work, we need to take BPS bounds given by combinations of chargesin the Cartan subalgebra of u (2 , | . Among the previous set, they are given by the rotationones S = 12 (cid:16) a − a + b − b (cid:17) , S = 12 (cid:16) − a + a + b − b (cid:17) , (D.5)and of the su (4) Cartan charges J = 12 (cid:16) − c − c + c + c (cid:17) , J = 12 (cid:16) − c + c − c + c (cid:17) , J = 12 (cid:16) c − c − c + c (cid:17) . (D.6)In addition, the u (2 , |
4) algebra contains three u (1) charges: the bare dilatation operator D , the central charge C and the hypercharge B , given by D = 1 + 12 (cid:16) a + a + b + b (cid:17) , (D.7) C = 1 − (cid:16) − a − a + b + b − c − c − c − c (cid:17) , (D.8) B = 12 ( a + a − b − b ) . (D.9)All the letters of N = 4 SYM satisfy C = 0 , then they correspond to a representation of psu (2 , | . The complete algebra also contains the generators for translations and boosts P α ˙ β = a † α b † ˙ β , K α ˙ β = a α b ˙ β , (D.10)and the fermionic generators for supersymmetry plus the superconformal partners: Q aα = a † α c a , ˙ Q ˙ αa = b † ˙ α c † a , (D.11) S αa = c † a a α , ˙ S ˙ αa = b ˙ α c a . (D.12)Among the entire set of generators, an important role is played by the su (1 ,
1) subalgebraspanned by L = 12 (1 + a + b ) , L + = a † b † , L − = a b , (D.13)which is common to all the near-BPS limits considered in Section 2. They satisfy the com-mutation relations [ L , L ± ] = ± L ± , [ L − , L + ] = 2 L (D.14) In this Appendix, we call J i the su (4) Cartan generators instead of the notation Q i used in the main textto avoid confusion with the supercharges. N = 4 SYM are composed by the bosonic and fermionic fields listed in Table2, 3 and 4, plus the gauge fields strengths and the covariant derivatives. Schematically, theyare given by Φ : ( c † ) | i , χ : a † c † | i , ¯ χ : b † ( c † ) | i , (D.15) F : ( a † ) | i , ¯ F : ( b † ) ( c † ) | i , d : a † b † | i . (D.16)Normalization factors are omitted, while the precise labelling of the indices depends from thespecific letter; this can be easily found by considering the Cartan generators and the fieldssurviving the various near-BPS limits. In this Appendix we focus on the main cases consideredin this paper: the su (1 , |
1) and psu (1 , |
2) algebras. su ( , | ) algebra The BPS limit in the SU(1 , |
1) sector reads D − (cid:18) S + J + 12 J + 12 J (cid:19) = 0 , (D.17)which implies a = b = 0 , c = c = 0 , c = 1 . (D.18)Moreover, the vanishing of the central charge gives the additional condition c = 1 − a + b . (D.19)The letters in this sector are | d n Z i = 1 n ! ( a † b † ) n c † c † | i , | d n χ i = 1 p n !( n + 1)! ( a † b † ) n a † c † | i , (D.20)where the factors are chosen to achieve unity normalization: h d m Z | d n Z i = δ mn , h d m χ | d n χ i = δ mn . (D.21)The generators of the algebra are the following: there are four bosonic generators, generatingthe algebra su (1 , × u (1) , given by the set (D.13) plus the additional u (1) generator R = 12 c . (D.22)Then we have four fermionic generators, that we collect using the notation Q = a c † , Q † = a † c , S = b c , S † = b † c † . (D.23)The generators of this sector satisfy the following commutation relations: { Q, Q † } = L + R , { S, S † } = L − R , { S † , Q † } = L + , { S, Q } = L − , (D.24)[ L , Q ] = − Q , [ L , Q † ] = Q † , [ L , S ] = − S , [ L , S † ] = S † , (D.25)[ Q, L + ] = S † , [ Q, L − ] = 0 , [ Q † , L + ] = 0 , [ Q † , L − ] = − S , (D.26)[
S, L + ] = Q † , [ S, L − ] = 0 , [ S † , L + ] = 0 , [ S † , L − ] = − Q . (D.27)58 typical feature of supersymmetric-invariant theories is that the anticommutator of thesupercharges closes on the free Hamiltonian. Looking at Eq. (D.24), this points towards theidentification H = L + R = S + J . (D.28)On the other hand, we remark in Section 2 that a more natural choice for all the near-BPSlimits is to take L to be the free part of the Hamiltonian, see e.g. Eq. (2.75). Indeed,following the near-BPS limit (2.46), the free Hamiltonian of the system would naturally be S + J + 12 J + 12 J = L + 12 , (D.29)and then the choice of take instead L to represent the free part simply corresponds to aconvenent mass shift.In order to follow this interpretation, we introduce a linear combination of the originalsupercharges which closes on L instead of the combination S + J . We define Q = 1 √ Q + S ) , Q † = 1 √ Q † + S † ) . (D.30)Since { Q, S † } = 0 , { Q † , S } = 0 , (D.31)we obtain {Q , Q † } = L = (cid:18) S + J + 12 J + 12 J (cid:19) − . (D.32)This representation of the supercharges is used in Section 2 and to build the superfield for-mulation in Section 4. psu ( , | ) algebra The BPS limit in this sector reads D − ( S + J + J ) = 0 , (D.33)which implies a = b = 0 , c = 0 , c = 1 . (D.34)The vanishing of the central charge in the representation gives c + c = 1 − a + b . (D.35)The set of letters of the sector is | d n Z i = n ! ( a † b † ) n c † c † | i , | d n χ i = √ n !( n +1)! ( a † b † ) n a † c † | i , (D.36) | d n X i = n ! ( a † b † ) n c † c † | i , | d n ¯ χ i = √ n !( n +1)! ( a † b † ) n b † c † c † c † | i , (D.37)where the prefactors ensure a unit normaliztion h d m Z | d n Z i = δ mn , h d m χ | d n χ i = δ mn , (D.38) h d m X | d n X i = δ mn , h d m ¯ χ | d n ¯ χ i = δ mn . (D.39) Notice that we are choosing conventions such that the fermion field ψ n creates the state −| ¯ χ i . su (1 ,
1) subalgebra are the same as in the su (1 , |
1) sector. TheR-symmetry generators form now a su (2) subalgebra, given by R = c † c , R = c † c , R = 12 ( c − c ) . (D.40)The fermionic generators can be collected in the convenient basis Q = a c † , Q † = a † c , S = b c , S † = b † c † , (D.41)˜ Q = a c † , ˜ Q † = a † c , ˜ S = b c , ˜ S † = b † c † . (D.42)They satisfy the following commutation relations: { Q, Q † } = L + R , { S, S † } = L − R , (D.43) { ˜ Q, ˜ Q † } = L − R , { ˜ S, ˜ S † } = L + R , (D.44) { Q, ˜ Q † } = R , { S, ˜ S † } = − R , { ˜ Q, Q † } = R , { ˜ S, S † } = − R . (D.45)Similarly to the su (1 , |
1) sector, we would like to identify the free part of the Hamiltonianto be L , and define a linear combination of the supercharges such that they close on thisgenerator. This is also motivated by the fact that L differs by the combinations of Cartancharges defining the PSU(1 , |
2) by a constant: S + J + J = L + 12 . (D.46)We then define Q = √ ( Q + S ) , Q = √ (cid:16) ˜ Q + ˜ S (cid:17) , (D.47) Q † = √ (cid:16) Q † + S † (cid:17) , Q † = √ (cid:16) ˜ Q † + ˜ S † (cid:17) , (D.48)which satisfy {Q , Q † } = {Q , Q † } = L , (D.49) {Q , Q † } = (cid:0) R − R (cid:1) , (D.50) {Q , Q } = {Q † , Q † } = 0 . (D.51)They correspond to the supercharges defined in (2.109). References [1] N. Beisert et al. , “Review of AdS/CFT Integrability: An Overview,”
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