PPrepared for submission to JHEP
Bosonization in Non-Relativistic CFTs
Carl Turner a a Department of Applied Mathematics and Theoretical Physics,University of Cambridge,Cambridge, CB3 OWA, UK
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Abstract:
We demonstrate explicitly the correspondence between all protected operatorsin a 2+1 dimensional non-supersymmetric bosonization duality in the non-relativistic limit.Roughly speaking we consider SU ( N ) Chern-Simons field theory at level k with N f flavoursof fundamental boson, and match its chiral sector to that of a SU ( k ) theory at level N with N f fundamental fermions. We present the matching at the level of indices and individualoperators, seeing the mechanism of failure for N f > N , and point out that the non-relativistic setting is a particularly friendly setting for studying interesting questions aboutsuch dualities. a r X i v : . [ h e p - t h ] D ec ontents N f > N Versus N f = N – 1 – Introduction
One of the highlights of recent work in the theoretical physics community has been asignificant leap in our understanding of field theoretic dualities, especially in 3 dimensions.Part of this is a new appreciation for old ideas from condensed matter physics suggestingrelations between theories of bosons and fermions [1–4] and particle-vortex duality [5, 6] –and newer ideas about fermionic dualities [7, 8] – and part of it has its origins in large N physics [9–11]. These have been united [12–14] (with supersymmetry [15–20] as a guidingprinciple, and level-rank duality [21] as the algebraic heart) into a coherent set of precisestatements about the equality of partition functions for (roughly speaking) SU ( N ) k bosonictheories and SU ( k ) N fermionic theories, forming an elaborate web of dualities [22, 23].Meanwhile, in [24, 25], the spectrum of a broad class of non-relativistic Chern-SimonsCFTs was analyzed. In particular, by exploiting a close relationship with a superconformaltheory, the exact dimensions of a set of protected operators were calculated.But, as discussed in [26], these non-relativistic CFTs inherit certain dualities fromtheir relativistic parents. In particular, the bosonization dualities proposed in [13] (andgeneralized slightly in [26]) have well-defined non-relativistic limits offering exact corre-spondences between non-relativistic Chern-Simons-matter theories coupled to bosons andfermions. Of course, non-relativistic field theory is essentially quantum mechanics, andthese dualities are fancier versions of flux attachment [27]. Yet the nature of bosonizationremains non-trivial even in this simpler setting.In fact, the chiral physics we discuss is intimately related to the WZW model – indeed,see [28] for a direct map between a different phase of this theory and the WZW model –and the 1+1 dimensional non-Abelian bosonization story [29].In this paper, we will explain how bosonization is realized in the setting of non-relativistic Chern-Simons-matter field theories. We begin in this introduction with anoverview of the results of this paper. Then in section 2 we give clear statements of thepairs of theories which are dual in this sense, and recapitulate the relevant results of [24, 25].This done, in section 3 we will present the correspondence between the protected stateswhose dimensions are known exactly in these theories. In section 4 we will discuss thismatching and its limitations. Finally, in section 5 we will conclude with a brief summary.
Key Results
The bosonization dualities discussed above generically look like a SU ( N ) k ↔ SU ( k ) N level-rank duality with some extra structure. We will demonstrate explicitly the non-supersymmetric duality between the following two Chern-Simons-matter theories: Bosonic Theory: U ( N ) k,k + nN coupled to N f fundamental non-relativistic scalars. Fermionic Theory: U ( k ) − N, − N coupled to N f fundamental non-relativistic fermionsand, through a BF coupling, to U (1) n . – 2 –e will explain various subtleties here (especially regarding the U (1) parts of thisduality) in section 2. For now, we merely remark that the first and second subscripts oneach gauge group indicate the non-Abelian and Abelian Chern-Simons levels respectively.Importantly, we will find that the duality holds only for N f ≤ N .Both theories will exhibit non-relativistic conformal symmetry [30–32], associated toan extra SO (2 ,
1) symmetry beyond the naive Galilean group. This is very analogous tothe more familiar relativistic conformal symmetry, with features such as a state-operatormap, though it is generally simpler to understand. In particular, in previous work thedimensions of a protected class of chiral operators were calculated exactly [24, 25].We will find that we can give a precise mapping between the chiral operators of boththeories. As we shall see and explain, this even includes operators transforming non-trivially under the global part of the gauge group. The mapping takes the form of a simpleexplicit recipe to construct the dual of any given operator. There is a close relation to theworld of 1+1 dimensional level-rank duality of conformal field theory, which we will brieflyoutline.It can also, of course, be encoded as an equality of certain indices between the twotheories. We will describe how to construct these indices by starting with the spectrum offree particles and implementing the flux attachment. The mapping and indices are givenin section 3.In order to make this duality fly, we have to understand an important subtlety ofChern-Simons theory: not all operators one might naively write down are permitted inthe spectrum. This is dictated by the fusion rules of the underlying affine Lie algebra.Normally, when one brings together p particles transforming in representations R i of somegroup, they collectively transform in arbitrary irreps in the tensor product R ⊗ R ⊗· · ·⊗ R p .However, the fusion rules knock out certain terms in that decomposition into irreps. Inparticular, in the language of Young diagrams, no representation of SU ( N ) k with morethan k columns is permitted.Understanding precisely how these are implemented has important, and slightly subtle,ramifications for our spectra – we will address this fully only in section 4 as we go throughseveral simple examples of the duality, understanding how fusion rules are implementedand then investigating how the number of flavours affects the duality. We will find that theduality stated holds only for N f ≤ N , as the fermionic theory given above contains extrastates.One nice perspective on this duality is that it is a return to the world bosonizationoriginally came from: non-relativistic flux attachment. When we are done, we will havea detailed and very down-to-earth understanding of exactly how attaching flux to bosonsand fermions works (including how this is restricted by fusion rules, and what it looks likeat the level of partition functions or indices) and exactly how this gives rise to dualities.Such a simple framework makes for an excellent testbed for investigating more potentialdualities, and we leave this open for further work.– 3 – Duality of Non-Relativistic Conformal Theories
Non-relativistic theories, and especially those with conformal symmetry, are not particu-larly familiar to many, so in this section we will summarize some of their key properties.We will begin by describing their Lagrangians, before taking a moment to discuss someof the subtleties of the duality as relevant for these theories. Then we will recapitulateresults about their spectra. This will leave us ready to begin explaining how the duality isimplemented in section 3.
Recall that we are interested in the duality U ( N ) k,k + nN + N f scalars ←→ U ( k ) − N, − N × U (1) n + N f fermions (2.1)where both theories are non-relativistic and have conformal symmetry.In order to specify these theories completely, we specify their gauge and matter sectorsseparately. Firstly, we should explain the notation U ( N ) κ,κ (cid:48) . This refers to the gaugegroup U ( N ) κ,κ (cid:48) = U (1) κ (cid:48) N × SU ( N ) κ Z N (2.2)with the corresponding Lagrangian L CS = − κ π tr (cid:15) µνρ (cid:18) a µ ∂ ν a ρ − i a µ a ν a ρ (cid:19) − κ (cid:48) N π (cid:15) µνρ ˜ a µ ∂ ν ˜ a ρ where a is the SU ( N ) part of the gauge field and ˜ a is the U (1) part. Notice that thediscrete quotient of (2.2) enforces the constraint κ (cid:48) − κ ∈ N Z . Indeed, the bosonic theoryexhibits the general solution of this constraint, using the parameter n .For the particular values of n = 0 , ± , ∞ it reduces to special cases analyzed in [13, 14]which are dual to fermionic theories with a single gauge field. These are realized as follows: n = 1 : U ( N ) k,k + N + N f scalars ←→ U ( k ) − N, − N − k + N f fermions n = − U ( N ) k,k − N + N f scalars ←→ U ( k ) − N, − N + k + N f fermions n = 0 : U ( N ) k,k + N f scalars ←→ SU ( k ) − N + N f fermions n = ∞ : SU ( N ) k + N f scalars ←→ U ( k ) − N, − N + N f fermionsFor general n , the fermionic theory requires an extra U (1) n factor which strictly shouldnot be integrated out. This is coupled in via a so-called BF term, which couples the overall U (1) ⊂ U ( k ) field ˜ a to the extra U (1) n field b via the simple term L BF = k π (cid:15) µνρ ˜ a µ ∂ ν b ρ .Nonetheless, these are all easily related to each other by gauging U (1) symmetries andintegrating out fields, so in the following we will mainly find it simplest to work with the– 4 – = 1 case in which the Abelian and (renormalized) non-Abelian levels come out to be thesame.All that remains is to describe the matter couplings. These are simply given by L bosonic = iφ † i D φ i − m (cid:126) D φ † i · (cid:126) D φ i − m φ † i f φ i L fermionic = iψ † i D ψ i − m (cid:126) D ψ † i · (cid:126) D ψ i − m ψ † i f ψ i where i = 1 , . . . , N f is a flavour index, D = d − ia − i ˜ a denotes the appropriate covariantderivative, and f is the field strength associated to the connection a +˜ a . We have suppressedgauge indices. It is helpful to see the form of Gauss’s laws, the equations of motion for a and ˜ a , forsuch a theory. Looking at just the Abelian sector of the bosonic theory for an example, wefind tr f = 2 πk + nN φ † i φ i . (2.3)Hence the last term in each action is essentially a quartic self-interaction (equivalently, atwo-body delta-function interaction) for the matter fields with a very special value. Asdiscussed in [24, 25] following [33, 34], this is a natural choice in a renormalization groupsense (not too surprising given the theory is conformal here). It is also special in thatthere is a supersymmetric completion of such Lagrangians; similarly, this is the Bogomolnypoint.This is all reflected in the fact that one can rewrite the above terms in a very neatway, leading to the Hamiltonians H bosonic = (cid:90) d x m |D z φ i | H fermionic = (cid:90) d x m |D z ψ i | . (2.4) Bosonization dualities in non-Abelian theories are essentially level-rank dualities of theunderlying gauge groups, so we expect that, loosely speaking, a SU ( N ) k theory with bosonsshould be dual to a SU ( k ) N theory with fermions. However, there are several subtletieswhich have been addressed to arrive at (2.1):1. The level-rank duality does not deal with the U (1) parts of the dual theories. In fact,as first clarified in [13], one always needs to include non-trivial U (1) gauge groups inimplementing these dualities.2. It is possible that the implementation of bosonization may involve a discrete symme-try: parity. This is in fact the case – indeed, the SU ( N ) k theory is dual to somethinglike SU ( k ) − N (see, for instance, [14]). Note that we make a different sign choice for the last Pauli-type term to that originally made in [24].This is to ensure that we get the theory with the correct parity. – 5 –. With the introduction of N f fermionic fields on one side of the duality, there isa renormalization of the Chern-Simons level. In the relativistic case, this shift isby ± N f / ± N f /
5. Finally, there is a further possible renormalization of the non-Abelian Chern-Simonslevel according to the choice of regularization for loop integrals in the field the-ory. This comes from gluon loops. The usual approach is Yang-Mills regularization,in which a small kinetic term is added to the photon and then decoupled; in thisapproach, one finds that quantities in the SU ( N ) k theory depend not on k butˆ k = k + N sgn( k ). The results of [24, 25] were presented in another regularizationscheme (equivalent to dimensional regularization) in which there is no such shift.Therefore in importing results from there, we include this shift.6. It may be that this duality does not hold for all numbers of flavours N f . We will seethat we require N f ≤ N for it to work out in our case, where N is the number ofcolours in the bosonic theory. The above theories both have the interesting property that a part of their spectrum isprotected. (In the supersymmetric completion, this is the BPS portion of the spectrum.)Thinking of these theories as specifying the quantum mechanics of anyonic particles, if oneadds a harmonic trap, one finds a class of states called linear states whose energy dependslinearly on the inverse Chern-Simons level. Alternatively, thinking of them as defininga CFT with an appropriate algebra of local operators, one finds that the dimensions of acertain class of operator depend on the level in the same way. These pictures are related bya state-operator map, which (unlike in the case of relativistic CFTs) simply maps operatorsin the original theory to states in the theory with a harmonic trap added. This part of thespectrum was analyzed in detail in [24, 25].One part of those papers was to understand what the operators of the CFT are. Asa consequence of Gauss’s laws, we are obliged to turn on the gauge field when turning onthe matter fields – indeed, in the example of (2.3), it is clear that excitations created by φ † carry magnetic charge. In [26], these two contributions added together. The reason for the distinction is that we make adifferent choice of how to integrate out the fermions. Presumably this is a distinction associated to thechoice of ground state of the theory; here we are interested in the conformal phase. – 6 –he result is that the physical excitations of each system are best expressed in terms ofdressed versions of the φ operators, carrying magnetic charge. But there is a simple way toachieve this if we sit on the plane. We simply dress φ with a Wilson line stretching out toinfinity. This, being gauge invariant except under transformations at infinity, automaticallysatisfies Gauss’s law. Hence we defineΦ i ( x ) = P exp (cid:18) i (cid:90) x ∞ a + ˜ a (cid:19) φ i ( x ) (2.5)and similarly Ψ, though for the latter we need to include an extra Wilson line for the extragauge field, and adjust the charges to satisfy both Gauss laws. Path-ordered exponentialscan also be seen in solutions to the Knizhnik-Zamolodchikov equations which arise naturallyin anyon quantum mechanics; indeed, this is what imbues the particles with their anomalousstatistics [36].It is not immediately obvious that we can think of these as analogous to the local operators of a relativistic CFT, due to the manifestly non-local Wilson line, but in fact onecan sensibly discuss things like the scaling dimension of such an operator, and arguments tothat effect are given in [25]. (One key reason for this is the difference in the state-operatormap. In the relativistic case, the states live on a sphere, and this means we have nowhere torun the Wilson lines to. The non-relativistic harmonic trap has no such issue.) This meansthat these Φ fields are useful objects to discuss, though their extended nature means theyhave some significant subtleties (related to fusion rules, which constrain how they may becombined) to which we will return later.Since we are interested in operators at a point, from here on we will understandoperators to be evaluated at the origin, so Φ i = Φ i ( ) and so forth. Then, letting ∂ = ∂ z and ¯ ∂ = ∂ ¯ z , the operators we are interested in are linear combination of terms like O i ··· i p = p (cid:89) m =1 ∂ l m ¯ ∂ l (cid:48) m Φ † i m and similarly for Ψ in the fermionic theory. Note that these transform in some representa-tion R of the global part of the gauge group as well as the SU ( N f ) flavour symmetry, butwe have suppressed the indices of the former.Note also that operators which are total derivatives are (in a sense made precise interms of the conformal algebra in [24, 25]) descendants of simpler operators. We ignorethese in the following.Now one can easily express the protected operators in terms of the Φ i , Ψ i . We findthat they are simply those with only ∂ derivatives, so l (cid:48) m = 0. In the bosonic theory, theylook like this: O i ··· i p = p (cid:89) m =1 ∂ l m Φ † i m The dimension of such an operator is given by∆ = p − ( J − ps )– 7 –here J is its angular momentum, appropriately regularized by subtraction of each indi-vidual particle’s spin. One may think of this essentially as a measure of the binding energyof the composite operator. (This saturates a lower bound on the dimension of any operatorwith these eigenvalues of the number operator and angular momentum.)Moreover, there is an elegant formula for this angular momentum. There is clearlysome angular momentum associated to the derivatives ∂ l m in the above expression for O ,so let us account for this by defining J − ps = − p (cid:88) m =1 l m + J . (2.6)The more interesting thing is what remains. It is helpful to consider each factor of thegauge group separately. Suppose the gauge group is a product of the groups G I at levels k I . Then take O to be in a definite representation R I of each factor. Then J = − (cid:88) I C ( I )2 ( R I ) − p C ( I )2 (fund)2ˆ k I (2.7)where C ( · ) is the quadratic Casimir of a gauge group representation, defined by (cid:88) α t α [ R ] t α [ R ] = C ( R ) (2.8)for t α generators in the appropriate representation, normalized by tr t α t β = δ αβ . We havewritten the solution in terms of the renormalized level ˆ k I for each factor.The spin s of the individual particles is given by a related formula: s = − (cid:88) I C ( I )2 (fund)2ˆ k I − (cid:40) ‘fermionic’ theory (2.9)For example, consider an Abelian theory with ˆ k = 1. We see that bosons pick up ahalf-integer spin and fermions an integer spin, as expected.From this point of view, it is clear that something fairly non-trivial must happen tomatch these dimensions across the duality. This is what we will describe in section 3. The above formulae are on the one hand specializations of the results of [25] as those holdfor φ in arbitrary representations of the gauge group; although on the other hand, theyare also slight generalizations of those since here we include extra gauge group factors.However, we note also that these formulae are clearly intimately related to the so-called minimal energies of the affine Lie algebra representations associated to R I [37], or theconformal dimensions of the operators in WZW theory.The key idea is one we have already encountered, in fact, in (2.5). This, roughlyspeaking, takes the form of a decoupling transformation which separates the Chern-Simons– 8 –eld theory from the matter fields. Similar transformations are used in the free fermionconstruction of Wess-Zumino-Witten theories in 2 dimensions [38, 39]. They also makedirect contact with old ideas about the relationship of Chern-Simons theory on manifoldswith boundary to conformal field theory [40].Recall a z is flat away from φ insertions. Suppose there were no φ insertions. Then wecould write a z = ih − ( ∂ z h ) for some h , and ∂ z ( hφ ) = h∂ z φ + ( ∂ z h ) φ = hD z φ and hence one can write the action elegantly in terms of the fields ˜ φ = hφ and h , withthe Hamiltonian (2.4) becoming simply H ∝ (cid:82) d x | ∂ z ˜ φ | . Moreover, these two fields arecompletely decoupled, except for Gauss’s law, which would impose boundary conditions on h at each ˜ φ insertion. The path integral, in this sector, reduces to a Wess-Zumino-Wittentheory for h on the boundary at infinity. Of course, this is a very trivial observation withoutany matter insertions!As we add φ insertions, we start obtaining multiple boundary components, and wecannot typically expect a single-valued h . In general, this can result in a much morecomplicated path integral; indeed, even for Abelian anyon theories, not much is knownabout the dimension of generic operators. However, for the special case of chiral operatorswe obtain exactly the construction of 1+1 WZW models alluded to in the last section of[40], with a WZW field h decoupling from some ˜ φ source as above. Accordingly, we canunderstand these 3d chiral operators in terms of WZW chiral primaries, and this underliesthe nice formula (2.7) for the corresponding operator dimension.Of course, once we understand that there is a link between the chiral operators weare studying and operators in WZW models, we find ourselves in the world of level-rankdualities [21] and non-Abelian bosonization [29] – thus it is perhaps no surprise that wecan understand the duality of this sector of the 3d theory in terms of level-rank dualitiesin 2d. The goal of this paper, then, is to elucidate precisely what this duality looks like interms of the 3d fields.Finally, as mentioned in the introduction, we note that in [28] a precise relationshipwas established between the chiral sector of a deformation of this CFT and a WZW model.Specifically, a chemical potential was added, and the moduli space of v vortices in a har-monic trap was studied. The dynamics within this moduli space, as v → ∞ , was shown tobe precisely that of the chiral Wess-Zumino-Witten theory by matching both the algebraand the partition function of the former to the latter. This is clearly similar to what wehave seen here: the chiral sector of the theory matches a chiral WZW model.– 9 – A Recipe for Matching Protected Operators
We will describe the matching for protected operators in three stages. Firstly, we willobserve that the single-particle operators match. Secondly, we will explain how “small”multi-particle states, namely those which avoid triggering the fusion rules, also match.Finally, we will explore how arbitrary multi-particle states match.Before we embark on this path, it is worth emphasizing that we are going to provide amatching not only between gauge singlet operators, but actually between gauge-dependentoperators. This obviously is not a bijection between operators (consider the single-particleoperators in e.g. a U ( N ) ↔ U (1) duality), but it is a bijection between their gauge orbits.We offer two perspectives upon this: • The first the point of view is that of Chern-Simons theory. Let us consider puttingsome charges into a Chern-Simons field theory, and asking that the state transformsunder some particular representation of the gauge group at infinity. We can do thisby inserting Wilson lines at the appropriate points. Then, as famously discussed in[40], one can analyze the Hilbert space of possible Chern-Simons states which canexist given these charges. These spaces are the objects we match. • The second is completely equivalent, but phrased in condensed matter language. Bycoupling matter to a Chern-Simons gauge theory, we are really looking at theories of anyons . With each physically distinct representation of a group SU ( N ), we associatea single species of anyon. Thus in this language, our duality is an implementation ofthe level-rank duality of anyon species.We emphasize that it is known that, for instance, the fusion rules dictating how anyonspecies can fuse into new compound anyons are preserved across level-rank duality [41].This is essentially the same thing as the statement that the Hilbert spaces of Chern-Simons states agree across level-rank duality. Our goal in this paper is to give descriptionsof the operators which create such states, demonstrating that their dimensions and angularmomenta agree across the duality.Note that there are two ways to handle what happens at infinity. One option is tocompactify and think of inserting a single Wilson line there too. The other is to dress treatthe theory as having a boundary, and add a WZW theory containing operator insertionsat that boundary. In either case, the fusion rules also constrain what representation theinsertion (and therefore the bulk state) may transform in. The duality will match partitionfunctions up to the level-rank duality of this global representation.With that all said, let us begin by considering the situation with single-particle oper-ators. The only possible matching is O i = ∂ l Φ † i ←→ ˜ O i = ∂ l Ψ † i (3.1)– 10 –nd this indeed works out very neatly. The dimensions of both operators (which are free)match trivially. The more pleasing thing is that their angular momenta are equal. For thecase of n = 1, this is realized simply by the identity J Φ † = − N k + N ) = − k × − ( N + k ) −
12 = J Ψ † . (3.2)But what about multi-particle states, where the anomalous (binding-energy-like) an-gular momenta kick in? Suppose that you have a representation R of some unitary group U ( N ). Then we candescribe it efficiently using an unreduced Young diagram with λ i boxes in the i th row.Here, “unreduced” means we allow full columns of N boxes in order to keep track of thetotal U (1) charge too.As mentioned above, it is simplest and perfectly general to focus on the n = 1 case,since here the U (1) level and the renormalized SU ( N ) level agree.The corresponding anomalous angular momentum associated with this diagram can beevaluated by using the expression for the quadratic Casimir in terms of the inner productof weights (cid:104) λ, λ + 2 ρ (cid:105) . (Here, ρ is the sum of fundamental dominant weights.) One obtains J = − C ( R ) − p C (fund)2ˆ k = − (cid:80) i (cid:2) λ i ( λ i − − ( i − λ i (cid:3) ˆ k (3.3)where we have written this last term in a very particular way, for reasons that will nowbecome clear.The protected operators we have discussed can be constructed by taking an operatorbuilt from the right number of bosons and derivatives, and contracting its gauge indices a m with a tensor M transforming in the representation R : O i ··· i p = M a ··· a p ∂ l Φ † a i · · · ∂ l p Φ † a p i p (3.4)We seek a dual for this operator, constructed from fermions, which transforms in the same SU ( N f ) representation. Let us consider operators of the form˜ O i ··· i p = ˜ M a ··· a p ∂ l Ψ † a i · · · ∂ l p Ψ † a p i p (3.5)Replacing bosons with fermions means that everything which was symmetrized is now anti-symmetrized and vice versa. So if we want the same SU ( N f ) representation, we must put˜ M in a representation ˜ R described by a partition λ T : λ = ←→ λ T = (3.6)– 11 –his, of course, is not a surprise: level-rank duality generally relates Young diagrams totheir transpositions, up to the important issues to be discussed in section 3.2.What is the anomalous angular momentum J of this state? It is easy to work outby looking at (3.3) the right way. The first term in the sum counts the number of pairsof boxes lying in the same row of the Young diagram. The second similarly counts thenumber of boxes lying in the same column. But upon transposition, these two terms aresimply interchanged.Moreover, recall that under the n = 1 duality, ˆ k → − ˆ N , but ˆ k = ˆ N = k + N whenboth quantities are positive. Hence in fact, J → J is invariant for these theories!This is the first clear manifestation of the multi-particle bosonization duality: theanomalous angular momentum for an bosonic operator and its naive dual (3.5) are identical.It follows that their dimensions ∆ and angular momenta J are also equal. This resultessentially reproduces old observations about the behaviour of Wilson line observablesunder level-rank duality [21].We see that given a λ describing a valid representation of SU ( N ) then the transpose λ T is a valid representation of SU ( k ) only if it has at most k columns. But this (neglecting SU ( N ) singlets for the moment) is precisely the requirement that the representation is an integrable representation of SU ( N ) k .This notion will be important for us. In general, when we bring operators transformingunder a gauge group with a finite level, rather than transforming in the tensor product,the composite operator transforms only in the representations specified by the fusion rulesof that theory [41], as mentioned in the introduction. In particular, SU ( k ) representa-tions whose reduced Young diagrams exceed k columns are always projected out. A nicecharacterization of the fusion rules is that when one brings together a fundamental weight(a partition which is a column of boxes) with some other integrable partition, one simplytakes the normal tensor product but then sets to zero any partition whose Young diagramis not integrable.This is the key subtlety associated with the procedure for attaching Wilson lines wepostulated above. Thinking of the physical states as being given by Chern-Simons theorydefined around fixed point charges, the Hilbert space of gauge configurations is only non-empty when we obey the fusion rules, as in the story told in [40].In summary, if we start with a λ describing a valid representation of SU ( N ) k , with nosinglet factors, then the transpose λ T is always a valid representation of SU ( k ) N . But wedo not have a procedure which works if we add many singlets to the operator. The above is clearly is only a part of the story. In general, there is no operator of theform (3.5), because if we include several SU ( N ) singlets in λ , then there may be too manyantisymmetrizations in λ T for it to be a valid representation of SU ( k ). But in order for the– 12 –ormulae for angular momenta and dimensions to be equal, we must include these singletsto get the correct U (1) contributions.From a more mathematical point of view, there is no bijection between representationsof affine special unitary Lie algebras. The usual level-rank duality picture only specifies abijection between representations of SU ( N ) k and SU ( k ) N modulo outer automorphisms .The outer automorphism groups are respectively Z N and Z k , and are associated with thecenter of each original group.The action of Z k upon reduced representations of SU ( k ) at level N is easy to describe:the generator σ adds a row of N boxes to the top of the diagram and then reduces it. (It iseasy to verify σ k = 1.) But from the point of view of the bosonic side of the duality, clearlythis looks much like adding a column of N boxes – that is, adding a singlet – before thenperforming a rather mysterious operation, namely removing complete rows of k boxes. This all suggests that the U (1) part of the gauge group must mix in some non-trivialway with some other source of angular momentum. This other source can only be deriva-tives.To see this, it makes sense to focus firstly on Abelian theories, where things are simpleand we only have these U (1) factors to worry about. Thus we will set N = k = N f =1, but keep n general. Consider the lowest-dimension protected operator in the bosonictheory: O = Φ † Φ † · · · Φ † (cid:124) (cid:123)(cid:122) (cid:125) p in U (1) n +1 (3.7)This should presumably match the lowest-dimension fermionic operator,˜ O = Ψ † ∂ Ψ † · · · ∂ p − Ψ † (cid:124) (cid:123)(cid:122) (cid:125) p in U (1) − × U (1) n ≡ U (1) − − /n (3.8)where we have integrated out the extra U (1) n for expedience, this being equivalent to doinga more careful calculation of dimensions. Indeed, this pans out very neatly:∆ O = p + p ( p − n + 1)∆ ˜ O = p + 12 p ( p − − np ( p − n + 1) = ∆ O This makes sense: the statistical parameters of these theories are θ = π/ ( n + 1) and˜ θ = − nπ/ ( n + 1) respectively, and they are related by θ = ˜ θ + π , or the distinction betweenbosons and fermions.Focussing on the n = 0 case for simplicity, this is the standard flux attachment proce-dure for turning bosons into free fermions. This can also be expressed as the identity q x ∂ x / ∞ (cid:89) m =0 − xq m = ∞ (cid:89) m =0 (1 + xq m ) (3.9) Note that the matching between singlets and maximally symmetric representations corresponds to thematching between baryons and monopoles discussed in e.g. [13]. – 13 –hich equates chiral indices for the two theories. (These are the partition functions ofthe chiral operators in the two theories, Z = Tr chiral [ x N q ∆ −N ], where N is the numberoperator. Each term in the product deals with m -derivative single particle operators, whilstthe prefactor on the left attaches flux to the bosons.) Writing ( x ; q ) n = (cid:81) n − l =0 (1 − xq k ) forthe q -Pochhammer symbol, we see that this identity combines two common special casesof the q -binomial theorem:1( x ; q ) ∞ = ∞ (cid:88) n =0 x n ( q ; q ) n and ∞ (cid:88) n =0 q n ( n − / ( q ; q ) n x n = ( − x ; q ) ∞ (3.10)The usual combinatorial interpretation of these results is precisely that partitions of aninteger n into integers and distinct integers are differentiated precisely by the inclusion of atriangular partition – in physics language, bosons and fermions are related by the inclusionof ∂ , ∂ , . . . , ∂ n − derivatives included in the chiral fermionic ground state.Here, we see clearly that indeed there is a mixing between explicit and anomalousangular momentum across the duality. This is the key idea we need to identify the matchingbetween operators.However, things are more subtle when we have a non-trivial SU ( N ) part to the gaugegroup. When the group sits at level k , we are forbidden from constructing operatorstransforming as (reduced) Young diagrams with more than k columns. From the point ofview of our unreduced Young diagrams, we may have more than k columns, but only byadding singlets (columns of N boxes) to the left of the diagram.From the point of view of the dual SU ( k ) N , of course, added singlets appear as extramaximal-length rows of N boxes at the top of the reduced Young diagram. Genericallythe diagram then needs reducing to be a SU ( k ) state. This is how we explore the orbit ofthe outer automorphism group under the duality.This means that, by tracking the total U (1) charge of the states transforming in theserepresentations, we actually obtain the outer automorphisms much more naturally than inthe above presentation. One simple way to understand why this is from the embedding u (1) Nk ⊕ su ( k ) N ⊕ su ( N ) k ⊂ u ( N k ) (3.11)commonly discussed in the context of level-rank duality. Consider the free fermion repre-sentation of the right-hand side; in fact, simply take a quantum mechanics of free fermions χ ia transforming in SU ( k ) × SU ( N ). The resulting states decompose into representationsof the left-hand side. One easily sees that things in the representation λ of SU ( N ) only ap-pear in representations SU ( k ) which are λ T or elements σ r ( λ T ) of its outer automorphismorbit, precisely because of the potential presence of singlets. But clearly the U (1) chargeprecisely counts the number of singlets, and hence disambiguates the level-rank map. The U (1) and SU ( k ) representation together specify the SU ( N ) representation uniquely; andsimilarly with k ↔ N .We need to check that this allows us to construct some sort of bijection which preservesthe angular momenta and dimensions. – 14 – onstructing a Matching Operator Suppose you are given a bosonic operator consisting of qk + r singlets and a part trans-forming under a reduced SU ( N ) representation µ which is integrable (i.e. with at most k columns). There are q blocks which are N × k rectangles, which upon transposition remaingauge singlets. Put these to the left of the new diagram. The remainder of the transposeddiagram – r rows of N blocks atop µ T – may be invalid if it has more than k rows. If itdoes, take the portion of the diagram below the k th row and place it instead at the topright of the diagram. This is necessarily a valid U ( k ) diagram, and also has at most N columns which are not singlets.An example of this process is depicted in Figure 1. Note also that the final SU ( k )representation is related to the transpose of the original SU ( N ) representation by the outerautomorphism σ r .We must now dress the operator with the appropriate derivatives so that a validfermionic object may be constructed transforming in this way. The recipe is simple enough:divide the completed diagram into sets of N columns labelled by I = 1 , , . . . . Then everyoperator sitting in the I th region must have I − I − N f = N = 3 and k = 2 and a bosonicoperator like O = (cid:16) Φ † [ a [ i Φ † a i Φ † a ] i ] (cid:17) (3.12)then the fermionic dual is O = (cid:16) Ψ † ( a [ i Ψ † a i Ψ † a ) i ] (cid:17) (cid:16) ∂ Ψ † ( b [ j ∂ Ψ † b j ∂ Ψ † b ) j ] (cid:17) (3.13)where each square contains an SU (2) antisymmetrization to make it a singlet. Here, thefirst squared bracket is labelled I = 1 and the second square is labelled I = 2, hence theadded derivatives.There are two questions to answer. Firstly, do the quantum numbers still match afterthis transformation? Secondly, does this really give a bijection between non-vanishingoperators? We will begin by proving that the answer to the first question is yes. Thesecond question proves more subtle, and will require some more work to understand. Proof that Quantum Numbers Match
We can think of the process depicted in Figure 1 as simply transposing and then movingeach N × k block alongside the previous one instead of lying on top of it. This is emphasizedby the labelling of these blocks as I = 1 , , . . . .We only ever move blocks of cells that are in columns containing a multiple of k othercells, and we only ever place them alongside rows containing a multiple of N other cells.– 15 – = 1 I = 2 I = 3 I = 4 I = 1 I = 2 I = 3 I = 4 (a) The U (4) → U (6) description Transpose σ σ (b) The corresponding SU (4) → SU (6) description Figure 1 : Transposing from U (4) ⊃ SU (4) to U (6) ⊃ SU (6) This guarantees that each cell we moves shifts J ∼ (column pairs − row pairs) / ( N + k )by an integer. Moreover, if we count pairs by summing for each cell the number of cellsabove it or to its left, that integer is easily seen to be ( I −
1) for each cell in the I th regionwhich we move. Constructing a corresponding index which matches for all these gauge-non-invariant objectsis now a little more subtle. It is illuminating (though not strictly necessary) to proceed byfirst starting with the index for free scalars or fermions, and then describing the effect offlux attachment on the index. – 16 –et us focus on the scalar theory, and for simplicity of the notation let us restrict tothe case of N f = 1 (though a straightforward generalization is available). Then with anon-Abelian U ( N ) symmetry, ungauged to begin with, we have Z free = N (cid:89) a =1 ∞ (cid:89) m =0 − ω a q m (3.14)where ω a are fugacities for the U ( N ) Cartan elements, so a = 1 , . . . , N . Expanding thisout, each power of ω a q m corresponds to a ∂ m Φ † a .Now the effects of the flux attachment are best understood in a basis of irreduciblerepresentations of the U ( N ). This means we want to decompose Z free into a sum overSchur polynomials, Z free = (cid:88) λ S λ ( ω ) f λ ( q ) (3.15)where we sum over all partitions λ with at most N parts, and S λ are the Schur polynomials S λ ( ω ) = (cid:88) σ ∈ S N σ ω λ ω λ · · · ω λ N N (cid:89) a>b (cid:16) − x i x j (cid:17) . (3.16)These precisely correspond to irreducible representations of U ( N ). The infinite series f λ ( q )describes the dimensions of all the chiral operators in this representation.There are two key effects of the flux attachment which we now include. Firstly, as wehave established, there is an anomalous dimension J [ λ ] associated with the gauging; in U ( N ) k,k + N this was J [ λ ] = − (cid:80) i (cid:2) λ i ( λ i − − ( i − λ i (cid:3) k + N for instance. This shifts S λ → q − J [ λ ] S λ in (3.15). Note that we implement this even fornon-integrable representations.Secondly, the fusion rules must be included. It might be rather surprising that thiscan be implemented effectively starting from the free spectrum given that most of therepresentations in (3.15) are not integrable and must drop out of the spectrum. In fact, ittranspires that this works out very elegantly. All we need a recipe for taking an arbitrary U ( N ) representation and spitting out the corresponding level k representation.It turns out that this is a standard procedure in the representation theory of affine Liealgebras. (See section 16.2.2 of [41].) The mathematical picture is that the weight space forthe affine Lie algebra is decomposed into affine Weyl chambers . One takes all non-integrableweights, and reflects them using Weyl reflections back into the affine fundamental chamber.This results in some new representation R ( k ) λ say, at the cost of r reflections. One mustthen include R ( k ) λ with the sign (cid:15) λ = ( − r in the fusion rules. Weights lying on theboundary of the fundamental chamber are left invariant by a reflection and so ‘contributewith both signs’; we take (cid:15) λ = 0. Even though some representations appear with minus– 17 –igns, they always cancel out other terms in the underlying tensor product, leaving simplya subset of the full tensor product. Putting this all together, we have the recipe S λ ( ω ) −→ (cid:15) λ q − J [ λ ] S R ( k ) λ ( ω ) (3.17)which can of course be implemented by integrating (3.15) against an appropriate kernelin a generalization of the usual approach for picking out gauge singlet operators, since (cid:104) S λ , S µ (cid:105) = δ λµ for an appropriately chosen inner product.Indeed, we can write the partition function for chiral states in the integrable represen-tation µ of U ( N ) as Z µ = 1 N ! (cid:32) N (cid:89) i =1 πi (cid:73) d ω a ω a (cid:33) (cid:89) a
It is productive to take this opportunity to explore the details of this matching in thesimplest possible settings. We will first look at the single flavour indices, and then continueto examine the operator matching in detail for larger N f . We will start by illustrating the rules about N f = 1 indices presented in section 3.3 in avery simple non-Abelian setting. Example: N = 2 , k = 1The simplest possible non-Abelian example is U (2) , with a single boson. Here, we findthat the free boson index looks like Z free = 1 + 11 − q Z (cid:48) free where we separate the singlet then factorize out total derivatives to leave only primaryoperators Z (cid:48) free = S + 11 − q S + q − q S + 1(1 − q )(1 − q ) S + q (1 − q )(1 − q ) S + 1(1 − q )(1 − q )(1 − q ) S + q (1 − q )(1 − q )(1 − q ) S + q (1 − q ) (1 − q ) S + · · · where we have listed all operators with up to 4 Φ insertions. Note that the denominatorof each term can be understood in terms of products of (1 − q h ) where h ranges over allhook lengths, and we have factored out the trivial 1 / (1 − q ) present in all diagrams. Thenumerators count the number of antisymmetrizations required (i.e. the minimum numberof derivatives needed to prevent the corresponding operator vanishing).Now for k = 1, many of these operators are affected by the fusion rules. One findsthat −→ −→ −−→ −−→
0– 19 –here we have not yet included the q − J terms.Hence the physical partition function, as a sum over all integrable partitions, butdropping the vacuum state and the total derivatives as we did for Z (cid:48) free , is as follows: Z (cid:48) = S + q × q − / − q S + (cid:18) q × q (1 − q )(1 − q ) − × q (1 − q )(1 − q ) (cid:19) S + (cid:18) q × q (1 − q ) (1 − q ) − × q (1 − q )(1 − q )(1 − q ) (cid:19) S + · · · = S + q / − q S + q (1 − q )(1 − q ) S + q (1 − q )(1 − q )(1 − q ) S + · · · But now consider the dual theory of U (1) − fermions. We find that˜ Z (cid:48) = S + q × q − / − q S + q × q − (1 − q )(1 − q ) S + q × q − (1 − q )(1 − q )(1 − q ) S + · · · and clearly we indeed see that Z (cid:48) ≡ ˜ Z (cid:48) in the sense of the identification (3.19). Example: N = 2 , k = 2By contrast, if we go to level 2, then the relevant fusion rules become −→ −→ − which leads to Z (cid:48) = S + q / − q S + q / − q S + q (1 − q )(1 − q ) S + q / (1 − q )(1 − q )(1 − q ) S + q (1 − q ) (1 − q ) S + · · · ≡ ˜ Z (cid:48) giving an elegant SU (2) ↔ SU (2) matching under which the second and third termsinterchange but all other displayed terms are left invariant. A General Lesson About Fusion Rules
There is some useful information about how fusion rules are implemented buried in theabove formulae.Let us first look at U (2) with k = 1 and consider p = 3 particle operators. The lowestdimension operator should be the operator with the fewest derivatives corresponding tothe only possible U (2) representation which for N f = 1 is naively O = Φ † [ a ∂ Φ † b ] Φ † c – 20 –ith dimension ∆ = 4. However, using our algorithm, the dual of this would be constructedas ˜ O = Ψ † ∂ Ψ † ∂ Ψ † which clearly vanishes. Thus either the correspondence is failing here, or these are not thecorrect operators to be looking at. Fortunately, the resolution is the less drastic secondoption.Let us look at our index for this theory and try to get some insight. We see that thecoefficient of S is shifted by the implementation of the fusion rules as q (1 − q )(1 − q ) −→ q (1 − q )(1 − q ) − q (1 − q )(1 − q )= q (1 − q )(1 − q ) (4.1)In particular, note that the first term in the series expansion is q – hence the lowestdimension operator in fact has two derivatives, as it must do to match the fermionic side.Moreover, the whole rest of the spectrum is shifted significantly. The interpretation of theshift from the q/ (1 − q ) → q / (1 − q ) suggests that a requirement to have “at least onederivative, and possibly more” is being replaced with a requirement to have “at least onepair of derivatives, and possibly more pairs”. Why is this?Consider the symmetrized operator Sym[Φ † a ( z )Φ † b ( z )Φ † c ( z )]. Then the fusion rulesinsist that the only non-vanishing components of this as z , z , z → z transform as only.This is what we implemented above. Suppose instead we start bringing z and z togetherfirst, so that the wavefunction becomes proportional to (cid:15) ab ; then for symmetry reasons,there must be some relative angular momentum, making the leading term proportional to( z − z ) (cid:15) ab . This corresponds to constructing the operator Φ † [ a ∂ Φ † b ] . But recall there is athird operator Φ † ( z ), symmetrized with the z , z terms. If ( z − z ) (cid:15) ab is to be the leading term as we bring z and z together, then there must be some more angular momentum.Put another way, the operator written above appears to be order O (( z − z ) ) and so bysymmetry it would also be O (( z − z ) ).Therefore, the first acceptable option can actually be written as O = Φ † [ a ∂ Φ † b ] ∂ Φ † c with the dual ˜ O = Ψ † ∂ Ψ † ∂ Ψ † and the duality works! Both operators have dimension ∆ = ˜∆ = 5.Moreover, the shifted excitation spectrum corresponds to the fact that any time wehave an extra derivative on the Φ † b term, we must also add one to Φ † c – such excitationsindeed come in pairs. This matches with the same requirement on the fermionic side forthe second two fermions. – 21 –his generalizes in a natural if sometimes subtle way; we must always include enoughderivatives to guarantee that any subset of operators may be brought together safely with-out violating the fusion rules. This means that if you take q ≤ p operators, where youmust take operators with fewer derivatives attached first, then it must be possible tochoose those q operators to obey the fusion rules. In the example above, consisting of theoperators Φ † a ∂ Φ † b Φ † c , we must first pick the Φ † a Φ † c terms, but these necessarily transformin the a, c symmetric representation, violating the k = 1 fusion rules. It turns out that there is something to say even about Abelian-Abelian dualities, so let usbegin there, before continuing as before and looking at theories where instead one of thetwo factors is Abelian.
Two Particle Examples
Consider U (1) with k = 1. Then for one flavour, the matching between (3.7) and (3.8) verystraightforwardly generalizes to a complete matching of the whole spectrum, as the indexidentity (3.9) demonstrates. But for N f = 2 something immediately goes wrong. Considerthe simple question: what is the bosonic dual of the fermionic operator ˜ O = Ψ † [ i Ψ † j ] ? Itis easy to verify that this (sticking to the n = 1 duality U (1) ↔ U (1) − ) has dimension˜∆ = 3 / O whichone can construct. In particular, the lowest dimension bosonic operator which is flavour-antisymmetric is O = Φ † [ i ∂ Φ † j ] with ∆ = 7 / too many flavours for the bosonic theory to support a sufficiently low-dimension operator to match thefermionic theory. This fits in with a general expectation: bosonization works whenever N f ≤ N , but not for N f > N [14].Let us try and generalize this issue. Suppose we consider instead U (1) bosons with k ≥
2, so that now the fermions carry an SU ( k ) spin but sit at level 1. N f = 1 : O = Φ † Φ † ˜ O = Ψ † [ a Ψ † b ] ∆ = ˜∆ = 2 + 13 = 73 N f ≥ O S = Φ † ( i Φ † j ) ˜ O S = Ψ † [ a ( i Ψ † b ] j ) ∆ = ˜∆ = 2 + 13 = 73 O A = Φ † [ i ∂ Φ † j ] ˜ O A = Ψ † [ a [ i ∂ Ψ † b ] j ] ∆ = ˜∆ = 2 + 1 + 13 = 103Here we have listed the lowest dimensional two-particle operators in both the symmetricand antisymmetric representations of SU ( N f ). The striking this here is that Ψ † ( a [ i Ψ † b ) j ] is nota valid contender for ˜ O A , since it transforms in a non-integrable representation of SU (2)at level 1. (This operator would have ˜∆ = 5 / U (2) bosons with k = 1, againwith 2 particles. Similarly to the above, we are forbidden bosonic states which are gauge– 22 –ymmetric, so two particles must always be in a gauge singlet. This means that the lowestdimension operators look a little different according to whether N f ≥ N f = 1 : O = Φ † [ a ∂ Φ † b ] ˜ O = Ψ † ∂ Ψ † ∆ = ˜∆ = 2 + 1 −
13 = 83 N f ≥ O S = Φ † [ a ( i ∂ Φ † b ] j ) ˜ O S = Ψ † ( i ∂ Ψ † j ) ∆ = ˜∆ = 2 + 1 −
13 = 83 O A = Φ † [ a [ i Φ † b ] j ] ˜ O A = Ψ † [ i Ψ † j ] ∆ = ˜∆ = 2 −
13 = 53But again, it seems as if there is no problem for arbitrarily large N f .Of course, we have only investigated the two-particle sector. It turns out that this isnot usually sufficient to explore the issue we are interested in. Three Particle Operators
Consider U (2) with k = 1 and look at p = 3 particle operators. We have already discussed N f = 1 in the context of the indices, and seen that everything works out nicely once weunderstand how to implement the fusion rules correctly.Now let us stick with N = 2 and k = 1 but move to N f = 2 and then N f ≥
3. We expectthat the former should work well but that the latter should break. Here are the lowest-dimensional operators transforming in the specified (unreduced) SU ( N f ) representations: N f = 2 : O = Φ † [ a ( i ∂ Φ † b ] j ∂ Φ † ck ) ˜ O = Ψ † ( i ∂ Ψ † j ∂ Ψ † k ) ∆ = ˜∆ = 5 O = Φ † [ a [ i Φ † b ] j ] Φ † ck ˜ O = Ψ † [ i Ψ † j ] ∂ Ψ † k ∆ = ˜∆ = 3 N f ≥ O = Φ † [ a ( i ∂ Φ † b ] j ∂ Φ † ck ) ˜ O = Ψ † ( i ∂ Ψ † j ∂ Ψ † k ) ∆ = ˜∆ = 5 O = Φ † [ a [ i Φ † b ] j ] Φ † ck ˜ O = Ψ † [ i Ψ † j ] ∂ Ψ † k ∆ = ˜∆ = 3 O = Φ † [ a [ i Φ † b ] j ∂ Φ † ck ] ˜ O = Ψ † [ i Ψ † j Ψ † k ] ∆ = 4 (cid:54) = ˜∆ = 2Here we see that, indeed, for N f ≥ N = 1, k = 2 and N f =2? With three particles, we find that the bosonic state of minimal dimension in therepresentation is O = Φ † [ i ∂ Φ † j ] Φ † k with ∆ = 5which is dual in our sense to ˜ O (cid:48) = Ψ † [ a [ i ∂ Ψ † b ] j ] ∂ Ψ † ck whereas in fact the lowest-dimension fermionic operator can be projected out from˜ O = Ψ † [ a ( i Ψ † b ] j ) ∂ Ψ † ck with ˜∆ = 4showing that, indeed, the duality is still violated.– 23 – .3 N f > N Versus N f = N We can easily find a violation by considering N f ≥ N + 1 for arbitrary k, N , and lookingat the lowest-dimensional operators transforming in the kN + 1flavour representation (where we have included the singlets to indicate this is an objectmade of k + N operators).Let us set aside the special case k = 1. (This is slightly different because there areno fusion rules in Abelian theories, but is straightforward to handle as in the examplesabove, and results in a larger discrepancy.) Then the corresponding minimal bosonic andfermionic operators can be projected out from O = Φ † [ a [ i · · · Φ † a N ] i N ] · Φ † ( a N +1 ( i N +1 · · · Φ † a N + k − i N + k − ∂ Φ † a N + k ) i N + k ) ˜ O = Ψ † [ a ( i · · · Ψ † a k ] i k ) · Ψ † ( a k +1 [ i k +1 · · · Ψ † a k + N − i k + N − ∂ Ψ † a k + N ) i k + N ] which indeed have different dimensions. To be concrete, their gauge representations anddimensions are, for the bosonic operator, O : = ⇒ ∆ = ( N + k ) + 1 + k ( k + 1) − N ( N − N + k ) = N + 3 k + 32 k + 1 N and for the fermionic operator˜ O : = ⇒ ˜∆ = ( N + k ) + 1 − N ( N + 1) − k ( k − N + k ) = N + 3 k + 12 N + 1 k – 24 –o that in particular ˜∆ = ∆ − N f > N is that the bosonic theoryis forced into using a derivative and gauge symmetrization to support the large flavourantisymmetrization. On the other hand, the fermion can take advantage of the largeflavour antisymmetrization to keep a relatively low dimension. Setting N f = N By contrast, at the special point N f = N everything is very nice. When the number offlavours and colours within the bosonic theory coincide in this way, the lowest-dimensiongauge-singlet operators take the form O r = (cid:16) Φ [ a [ i · · · Φ a N ] i N ] (cid:17) r (4.2)which have the property that in SU ( N ) k their dimension scales linearly with r ,∆ = (1 + 2 kN + N ) r N + k ) . (4.3)Their angular momentum is J = 0 – they are non-interacting, spinless bosons, decou-pling completely from the SU ( N ) Chern-Simons theory save for the constant shift to theirdimension.The operators O r are dual to fermionic operators roughly of the form˜ O r = (cid:16) Ψ ( a [ i · · · Ψ a N ) i N ] (cid:17) k (cid:16) ∂ Ψ ( b [ j · · · ∂ Ψ b N ) j N ] (cid:17) k · · · (cid:16) ∂ (cid:98) r/k (cid:99) Ψ ( c [ l · · · ∂ (cid:98) r/k (cid:99) Ψ c N ) l N ] (cid:17) r mod k (4.4)which are indeed also of matching minimal dimension in the U ( k ) − N theory. (Each k thpower is understood as having a gauge antisymmetrization. Note that one cannot startsymmetrizing the blocks of N together until one has formed up these sets of U ( k ) baryons.)Notice that the lowest-dimension bosonic operators break SU ( N ) flavour × SU ( N ) colour → SU ( N ) diagonal (4.5)in exactly the same way that BPS vortices do in the phase of this theory obtained byadding a chemical potential. (See [26, 42, 43] for a discussion of this theory from the pointof view of quantum Hall physics.)One reason that this is a particularly nice point is that there is, of course, a constrainedfermion realization of the affine Lie algebra of SU ( k ) − N consisting of a theory with N f = N flavours of fermion [38]. Our chiral states are in bijection with constrained fermion states,though in our picture their dimension is not due to a quartic rearrangement of their stress-energy tensor (via the Sugawara construction) but due to the flux attachment, and theproperties of the attached Chern-Simons Wilson lines. The difference in the approach isclear from the fact that N f and N are independent parameters for us.– 25 –onetheless, it is clear that N f = N is on algebraic grounds a very natural particularcase of the bosonization duality. Moreover, we note that all the N f < N dualities can ofcourse be obtained from the N f = N duality by simply restricting to sectors where one ormore flavours of boson or fermion are not turned on. In this sense, the N f = N case is thenatural parent for all these bosonization dualities. It would be nice to understand the whole picture of bosonization (beyond chiral states)more algebraically. It seems likely that attacking the N f = N duality is the right way todo this. We have seen that when the number of flavours N f is larger than N , the bosonizationduality necessarily fails. One perspective on the mechanical reason for this was that thelarge number of flavours allows us to construct fermionic operators which are suitablyantisymmetrized without using many derivatives; intuitively speaking, the fermions canget too close. The dimension of the resulting operator is too low to be dual to any bosonicoperator.There is a natural concern which might arise: can the fermions get too close? Specifi-cally, there is a unitarity bound [44] in these theories given by∆ , ˜∆ ≥ k < k > N f > N .This is reminiscent of similar situations in superconformal field theories in which therelationship between N f and N affects the IR fate of theories [45–47] – in particular, bothwhether or not the naive implementation of superconformal symmetry leads to unitarityviolations (which mean there are additional sectors of free operators), and whether or notcertain Seiberg-like dualities hold. We will comment on this at the end of this section. If one prefers to think in terms of integrating out fields, note one can choose to either integrate outfermions and bosons using one of two sign choices for the ‘mass’ term. One choice simply removes a flavouron both sides of the theory. The other choice additionally shifts the Chern-Simons level − N → − ( N −
1) onthe fermionic side and partially Higgses the gauge field on the bosonic side, breaking SU ( N ) → SU ( N − – 26 – belian Fermionic Theories, k = 1Here, although we have focussed on k >
0, it appears that including a large number offermion flavours can drive the dimension down towards the unitarity bound. Indeed, setting k = 1, consider the Abelian fermionic operator˜ O = Ψ † [ i Ψ † i · · · Ψ † i Nf ] . (4.7)The dimension of this operator, in U (1) (cid:96) , is˜∆ = N f − N f ( N f − (cid:96) . (4.8)Then we find that ˜∆ ≥ ⇐⇒ N f ≤ (cid:96) (4.9)and in particular the bound is violated for N f larger than this. It seems that a free operatorarises at N f = 2 (cid:96) , and then must decouple from the theory as N f grows beyond this point.But this is not necessarily the worst-behaved operator. Consider the sequence˜ O q = (cid:18) Ψ † [ i · · · Ψ † i Nf ] (cid:19) (cid:18) ∂ Ψ † [ i · · · ∂ Ψ † i Nf ] (cid:19) · · · (cid:18) ∂ q − Ψ † [ i · · · ∂ q − Ψ † i Nf ] (cid:19) (4.10)instead. Then we find that˜∆ = qN f + 12 N f q ( q − − qN f ( qN f − (cid:96) ≥ ∀ q ⇐⇒ N f ≤ (cid:96) . (4.11)More specifically, at the value N f = (cid:96) , this sequence has ˜∆ ∝ q , whilst away from this pointthe quadratic dependence on q has different signs for N f ≷ (cid:96) . Therefore, for any N f > (cid:96) there are infinitely many operators (for sufficiently large q ) which naively have increasinglynegative dimensions.The conclusion is that these operators are not being handled correctly. As describedin [24, 25], they correspond to non-normalizable states under the state-operator map, andshould be removed from the spectrum. But it is tempting to speculate that perhaps weshould also include some free, decoupled operators in the theory corresponding to all theseillegal states, much like the Jackiw-Pi vortices did in the bosonic k < U (1) level (cid:96) in the dualities, via the extra U (1) n factor from the introduction, though, it is not immediately clear that this phenomenon canbe at all related to the duality.However, suppose that we compute the dimension of the above operators for general n, N . Then the effective value of the statistical parameter 1 /(cid:96) is1 (cid:96) = nnN + 1 (4.12)– 27 –otice that if we focus on n ≥ k >
0) then this quantity ranges between 0 and 1 /N . It follows that, if we want ˜∆ ≥ q, n ≥ N f ≤ N . One reaches the same conclusion if one requires only that theoperators do not violate the bound at n = ∞ , where the duality is SU ( N ) k ↔ U ( k ) − N .(Note that for n <
0, it is much easier for operators to violate the unitarity bound;this is more reminiscent of the situation with Jackiw-Pi vortices, for bosonic theories with k <
0. We will not be interested in these sorts of violations.)This suggests that perhaps there is indeed a link between violations of the unitaritybound (for either positive n or simply the particular case n = ∞ ) and the duality. Non-Abelian Fermionic Theories, k > U ( k ) fermionic theories. If we considerthe combination of qN f baryons as a simple probe, and set n = ∞ for simplicity, we finda lowest dimension of˜∆ = qN f k + 12 q ( q − N f k − kqN f ( qN f − − qN F k ( k − N + k ) − qN f k ( qN f k − (cid:20) kN − k ( N + k ) (cid:21) and then because of the quadratic terms, once more˜∆ ≥ ∀ q ⇐⇒ N f ≤ N . (4.13)Once again, the boundary of validity of bosonization coincides with a unitarity bound.Notice that, on the bosonic side, we do not get these violations of the unitary boundfor our k > SU ( N ) k for definiteness. Only gauge antisymmetrizations contribute negatively to the anomalousdimension ∆, but we cannot construct operators with arbitrarily large gauge antisym-metrizations. In particular, we cannot construct large operators with quadratically nega-tive contributions to the anomalous dimension. Actually, for a single baryon (which is asbad as it gets) we see O = Φ † [ a [ i · · · Φ † a N ] i N ] = ⇒ ∆ = N − N − k + N ) > Good, Bad, Ugly
As we pointed out above, this picture is somewhat reminiscent of the story of “the good, thebad and the ugly” [46, 47]. Here, there is a Seiberg-like duality between two superconformaltheories which works only for certain values of N f relative to N . Moreover, this coincideswith whether the unitarity bound in the theory is (naively) violated or not.The “good” theories are those for which N f ≥ N , which interestingly is for sufficiently large numbers of flavours. Here, the naive implementation of the unitarity bound in the– 28 –R leads to no problems, and we also find 3d N = 4 SQCD infrared dualities of the form U ( N ) + N f hypermultiplets ←→ U ( N f − N ) + N f hypermultiplets .Meanwhile, a borderline “ugly” case at N f = 2 N − U ( N ) + (2 N −
1) hypermultiplets ←→ U ( N −
1) + (2 N −
1) hypermultiplets+1 free hypermultiplet .For N f < N −
1, outright violations of the bound occur, and no simple infrared duality likethe above holds. (In [47], a suggestion [48] for a duality in the “bad” cases N ≤ N f < N − N f ≤ N theories and (“bad”) N f > N theories. Our proposedfermionic dual is clearly somewhat subtle for large N f in that, proceeding naively, we canconstruct various disallowed operators. It is no surprise that it is in fact not the dual ofthe bosonic theory, which is well-behaved in this respect.Nonetheless, the fermionic theory does contain a subsystem of operators which are dual to the bosonic operators. (This is reminiscent of the partially successful “bad” dualsalluded to above.) This leaves open the possibility that there is some duality of U ( N ) k theory with N f bosons to a theory of fermions subject to some additional constraint whichprojects out the unwanted states; perhaps with the structure of a coset model.It would be interesting to explore this further.– 29 – Conclusion
We have described the non-relativistic bosonization duality U ( N ) k,k + nN + N f fund. scalars ←→ U ( k ) − N × U (1) n + N f fund. fermionsfor N f ≤ N , and how it fails for N f > N , by matching individual chiral operators in thesetheories, and seeing how the corresponding indices also agree. It would be nice to extendthis to non-chiral operators, for which explicit dimensions are not known, to try to gaininsight about their spectrum.We also noted that N f = N is a particularly natural point to study, since e.g. thereis a free fermion representation of the affine Lie algebra su ( k ) N in terms of N fermions,and since the other dualities follow from this one upon removing some flavours. It wouldbe interesting to try and understand the full spectrum at this point in terms of the affineLie algebra.We have seen that the bosonization fails precisely for the same theories as a unitaritybound on operator dimension is violated, in analogy with the story of supersymmetricSeiberg-like dualities (which require instead lower bounds like N f ≥ N ). We have alsospeculated that, if a mechanism which restricts which states are allowed in the fermionictheory is added, it might be possible to find a duality which does work for larger flavours.Perhaps this might take the form of a Seiberg duality whose supersymmetry has beenbroken.One lesson offered by this work is that since non-relativistic field theories can becontrolled as well as supersymmetric theories, with very little complication or excessive fieldcontent, they provide a simplified laboratory in which e.g. non-supersymmetric dualitiescan be probed. We emphasize, for example, that the restriction to fundamental fields inthis work is not necessary; the conformal dimensions for states built from operators in arbitrary representations are known (and are presented in the context of the relevant fieldtheories in [25]). It would clearly be interesting to use this approach to conjecture and testmore dualities. Acknowledgments
Thanks to Alec Barns-Graham for helpful discussions, and to Nima Doroud, David Tongand Nick Dorey for comments on the manuscript. CPT is supported by a Junior ResearchFellowship at Gonville & Caius College, Cambridge; much of the work was also completedwhilst supported by the European Research Council under the European Union’s SeventhFramework Programme (FP7/2007-2013), ERC grant agreement STG 279943, “StronglyCoupled Systems”. CPT is also grateful for hospitality from National Taiwan Universitywhilst the work was being finalized. This work has been partially supported by STFCconsolidated grant ST/P000681/1. – 30 – eferences [1] A. M. Polyakov,
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