Notes on index of quantum integrability

Jia Tian,Jue Houand Bin Chen,3

1 School of Physics and State Key Laboratory of Nuclear Physics and Technology,Peking University,No.5 Yiheyuan Rd,Beijing 100871,China

2 Center for High Energy Physics,Peking University,No.5 Yiheyuan Rd,Beijing 100871,China

3 Collaborative Innovation Center of Quantum Matter,No.5 Yiheyuan Rd,Beijing 100871,China

Abstract A quantum integrability index was proposed in Komatsu et al (2019 SciPost Phys.7 065).It systematizes the Goldschmidt and Witten’s operator counting argument (Goldschmidt and Witten 1980 Phys.Lett.B 91 392)by using the conformal symmetry.In this work we compute the quantum integrability indexes for the symmetric coset modelsandThe indexes of these theories are all non-positive except for the case ofMoreover we extend the analysis to the theories with fermions and consider a concrete theory:the CPN model coupled with a massless Dirac fermion.We find that the indexes for this class of models are non-positive as well.

Keywords: integrability,quantum index,coset models

1.Introduction

The study of integrability has a long history,which can date back to the time of the birth of Classical Mechanics4For a short history of integrability see [4]..However the understanding of integrability is far from completion,particularly in the context of quantum field theories (QFT).The classical aspects of integrable QFT are usually described by the Lax operator formalism,which allows us to construct local or non-local classically conserved charges.The quantum aspects5For reviews of integrable structure in QFT see for example [5].of integrable QFT are dictated by the S-matrix factorization and bootstrap [1].Integrability itself is noble while proving integrability is always involved with sophisticated guesses and conjectures.The seminal works of [2,3] show that the factorization of S-matrix is a consequence of the existence of higherspin quantum conserved currents.Nevertheless,the construction of quantum conserved currents is quite tricky,as the classical conserved currents are often anomalous at the quantum level.

In [6],Goldschmidt and Witten (GW) proposed a sufficient condition to prove the existence of quantum conserved currents.By enumerating all the possible local operators which can appear in the anomaly of the classical conservation laws one can tell whether there exist quantum conserved currents.Even though the GW argument is clear,the complexity in counting the possible local operators in practice by the brutalforce method goes wild quickly.Recently,Komatsu,Mahajan and Shao(KMS)[7]systematized the counting,and introduced a quantum integrability index I(J) for each spin J,which we call the KMS index,to characterize the existence of the quantum higher-spin conserved currents.It is a lower bound on the number of quantum conserved currents of spin J.If the KMS index I(J) is positive,it implies the existence of the quantum conserved currents of the spin J.One remarkable feature of the KMS index is that it is usually defined at the UV fixed point of the sigma-model,but it is invariant under conformal perturbation around a conformal field theory fixed point.This allows us to use the conformal symmetry to enumerate the gauge invariant operators according to their scaling dimensions in a systematical way such that the computation of the index is feasible.In [7],the indexes of the higher spin currents for the CPNmodel,the O(N)model and the flat sigma modelwere computed.

In this note,we would like to compute the KMS index for some other quantum integrable coset models,including the SU(N)/SO(N) model,the SO(2N)/SO(N) × SO(N) model and the CPNmodel coupled with a Dirac fermion.We find that the KMS indexes of higher spins in these models are all nonpositive except for I (4) in themodel.

The organization of the paper is as follows.In section 2,we review the GW argument and the KMS quantum integrable index.For a clear illustration we focus on a concrete example,O(N) model.In section 3,we compute the KMS index for the coset models SU(N)/SO(N)and SO(2N)/SO(N) × SO(N).These two models are conjectured to be quantum integrable.Also in section 3,we consider the models with fermions and show how to generalize the KMS index.We summarize our results in section 4.

2.GW argument and KMS index

In this section we briefly review the Goldschmidt and Witten’s arguments for quantum integrability [6] and the quantum integrability index,introduced by Komatsu,Mahajan and Shao[7].We will take the O(N) model to elaborate the analysis.

GW argument

In [6],Goldschmidt and Witten proposed a sufficient condition to diagnose the conservation of quantum higherspin currents in two dimensional sigma models.Their criterion is based on an operator counting analysis in sigma models.Consider a two dimensional sigma model with classical conserved current satisfying

Quantum mechanically,the classical symmetry may be broken such that the conservation equation is modified to

where the anomalous term A is a local operator with proper conformal dimension.However,if A can be written as a total derivative as

then one may redefine the current as

such that the redefined current is conserved quantum mechanically.The GW criterion is that if the number of A-type operators is less than the number of B-type operators then the quantum higher-spin current is conserved.As an example [6]we consider the O(N) σ model whose action is given by

The theory is classically conformal invariant,and it has conserved currents of even spin building from the stress tensor.The stress tensor of the theory isDue to the fact that?-T++=0,the currentsJn=(T++)nis conserved classically.Let us consider the classical conserved spin-4 currentand then (2.2) reads

To construct the A-type and B-type local operators,we frist fnid the building blocks,a list of fundamental independent local operators called the letters.The requirement that the operators should be O(N) invariant implies that the vector index of onemust contract with the one of anotherto get a O(N) singlet.Due to the constraint=1,we can claimand·are not in the list.On the other hand,the equation of motion(EOM)of the model is

which implies that the letters can not have cross derivatives.Therefore the possible letters are

with conformal dimensions

Since the conformal dimension of A isthe only possible A-type operators are

The conformal dimension ofB+andB-areh+=(4 ,0) andh-=(3 ,1) ,respectively.So they can be

It seems that there are five B-type operators,but that is not true because we have not imposed the EOM.In other words,these Btype operators are not independent,considering the EOM.To remove the redundancy we have to rewrite?±±Bin terms of A:

Therefore,there are only three independent B-type operators remaining after imposing the EOM.It implies that A can always be written as a total derivative so that the spin-4 current is conserved even at the quantum level.

KMS index

Following the GW argument,the authors in[7]proposed the index

In the brutal-force counting method,we have shown the most cumbersome step is to remove the redundancy in the counting of B-type operators,due to the on-shell equation of motion.Noticing that the difference A - B defines the set

Here IBP stands for the total derivative terms as known as Integration By Part.The setC can be interpreted as the set of local operators with proper quantum numbers after considering the EOM and IBP.This kind of object has a clear analogue in effective field theory (EFT) known as the operator bases[8].The crucial idea here is that as the index is invariant under conformal deformation,we can study the index at the UV fixed point where we can organize all the local operators with respect to the conformal multiplets schematically denoted as

As a result,the partition function Z for all the independent local operators (the letters) has an expansion with respect to the conformal group characterslabeled by the conformal dimension Δ and the spin j:

Applying the orthogonal property of the character,the KMS index (2.14) for the spin6The spin j has to be an integer in order to have an inversion formula.j could be computed by using an inversion formula [7]

Let us revisit the O(N) model with this approach.The single-letter characters corresponding to the letters (2.8) is

The multi-letter partition function is given by the plethystic exponential [8]:

To compute the quantum index I (4) we need the character

and the measure in the space (q,x)

Substituting into(2.14),one can find I (4) =1 which matches the results from brutal force method.We conclude this section by listing other KMS indexes for the O(N) model:

Thus,there also exists a spin-6 quantum conserved current,as predicted in [7].

3.Coset models

The sigma models on homogeneous spaces also known as symmetric coset models are important examples of classical integrable field theory7A recent review can be found in [9].For an integrable but not symmetric coset model see [10]..Applying the operator counting techniques developed for EFT [8],KMS proposed a systematic way to compute the integrability index for the coset sigma models,which are not necessary to be symmetric.

KMS index for cosetsConsider a coset G/H with the associated Lie algebra orthogonal decomposition

whereh andk represent the elements in subalgebra and coset,respectively.Introducing the left-invariant one-form

and its decomposition

the action of the sigma model can be written as

The coset model has the local symmetry:

and a global symmetry:

The local operators can be built fromg,kμ(x) and their covariant derivatives Dμwhich is defined as

By imposing EOM and the flatness condition of the leftinvariant one-form we can find the complete set of global Gsymmetry invariant letters

where the light-cone coordinates have been used.All the letters under the H gauge transformation transform ashkh-1.From g andkμ(x)we can built the Noether currents of the global G symmetry

Using the Noether currents we can find a set of H-symmetry invariant letters

In order to construct gauge invariant operators,KMS introduced auxiliary parameters which they call fugacities for the representations,and performed the Haar integration over the group H.As a result,the single-letter character is given by

and the multi-letter partition function is similarly given by the plethystic exponential

From the KMS index point of view,the quantum integrability is totally determined by the representation R and the measure dμ H.When the representation R is trivial i.e.χR= 1,the KMS index vanishes.It is not hard to verify this fact numerically.For example,the index for the spin-4 current is given by

It is obviously vanishing for the trivial representation.We can also understand it in an intuitive way.For any high-spin conserved currentthere exist a A-type operatorBecause no cross derivatives can appear there is no B-type operators then the KMS indexes have to vanish.But we want to stress that the vanishing of KMS indexes does not mean the theory is not integrable.Instead we should think that in this situation GW argument fails and in order to examine the quantum integrability we need some other tools or criteria

Let us revisit the O(N) model which can be viewed as the coset modelThe currents kμform a vector representation ofSO(N- 1) .For simplicity,we assumeN-1 to be even then the character of the vector representation is given by

and the Haar measure is given by

Using the formula (2.18),we find the following results:

The observation is that when N is small the integrability indexes depend on N but they become stable whenN≥ 7 and the stabilized values coincide with results (2.24).Our calculations(2.24)and(3.14)show that the two descriptions(2.5)and(3.4)of the O(N)model are only equivalent for large enough N.The discrepancy between two kinds of counting for small N is subtle.We believe that the counting in the coset description is reliable.The subtlety is that in the description (2.5) after imposing the constraints

The discrete symmetry plays an important role for the quantum integrability.For example,for the parity-symmetric theories,the existence of only one local higher-spin conserved current will guarantee the quantum integrability.For the models with discrete symmetry,the KMS indexes must be improved by imposing the discrete symmetry.In this case,we can modify the partition function by gauging the discrete symmetry groupas [7]

Imposing the discrete Z2charge-conjugation symmetry,the KMS indexes of the O(N) model become8In [7],the indexes I (4) ,I (6) and I (8) have been computed.

independent of N.Comparing with the results without imposing the discrete symmetry,we see that the indexes of spin 4 and 6 are always positive,and the indexes of higher spin are larger than the one without discrete symmetry.

In the next section,we will use this strategy to study a few classical integrable models.For coset models,the crucial step is to identify the representation of kμwith respect to the subgroup.That is involved with a representation decomposition problem.Since we only need the character of the representation we solve the problem in the following way.Firstly we separate the normalized generators{TM}of the group into the subgroup part{Ta} and the coset part{Tα}.Then we parameterize the subgroup element as

so that the representation R is given by

In the end we express the character of R in terms of the eigenvalues of h which are our auxiliary parameters of fugacities.

4.Applications

4.1.Cosets SU(N)/SO(N)

The exact S-matrices for the sigma models on the spaces SU(N)/SO(N) and SO(2N)/SO(N) × SO(N) were derived in[11] where the author also showed when the θ term equals π the sigma models have stable low-energy fixed points corresponding toSU(N)1andSO(2N)1Wess-Zumino-Witten models.The quantum integrability of these two models relies on the fact that non-local charges survive quantization[12].In this and next sections,we examine the conservation of local higher-spin currents using the KMS index.

To identify the generators of the subgroups SO(N)for the symmetric cosets SU(N)/SO(N) we can solve the following equations [13]

where Σ0ia an N × N complex symmetric matrix that satisfiesfor some complex number c.Using the Gell-Mann matrices as the generators of SU(3),one can find that

where we have normalized the generators as Tr [Tα Tβ]=δαβ.The character of the representation (3.22) is

Taking a higher dimensional analog of the defining generatorsλi,i= 1,… ,15.we fnid the decomposition of the normalized generators

The corresponding character of the representation(3.22)is given by

The observation is that the representation R is the totally symmetric representation[2 ,0...,0].Using the expressions the Haar measures for the groups SO(N) [8],we get

The negative indexes imply that the GW argument fails.

We now proceed to take care of the discrete symmetry.Imposing the charge conjugation discrete symmetry extends the gauge group from SO(N) to O(N).The orthogonal group O(N) consists of two connected components:O+(N)=SO(N)and the parity-odd componentO-(N).A general elementg-∈O-(N)is connected to an elementg+∈SO(N)through a parity transformation σ in the formg-=g+σ.For odd N,the parity transformation can be chosen to commute with the rotations due toO(2r+ 1) =SO(2r+ 1) ×Z2so thatσ[1]= -I.Noticingσ[2]= (-1)2Ianddμ-=dμ+we conclude that the Z2symmetry does not change the KMS index for odd N cases.For even N case,because ofthe parity transformation σ does not commute with the rotation anymore.The results9For example,see the appendix of [8].of the representation theory is that the general irreducible representation ofO(2r)are labeled byl= (l1,… ,l r) withl1≥ ...lr≥0,

with the corresponding characters

At the same time taking the measuredμ-= dμ Spone can find the KMS index with (2.18).In the example of N = 4,we obtain

In the end combining the two components with (3.19) gives total KMS indexes

So the KMS index does not predict the existence of the quantum conserved spin-4 currents or any other higher-spin currents for these coset models.This is actually true for other even N.In short,the high-spin KMS indexes for the cosets SU(N)/SO(N) are all negative for all N,no matter N is odd or even.

4.2.Cosets SO(2N)/SO(N) × SO(N)

The symmetric cosets SO(2N)/SO(N) × SO(N)are known as the Grassmannians.We present the details for the low-rank examples,and then conclude for general N.

Let us start with the lowest rank case

We will use the defining normalized generators for the orthogonal groups.In this case,the subgroup corresponds to the Cartan subgroup spanned by(T12,T34).The character of the representation Rabare

and the corresponding measure is

The product form of the character is due to the fact the coset is in the bi-fundamental representation:R= [ 1]1?[ 1]2.A direct calculation gives the KMS indexes

However the Grassmannian (4.12) is basically two copies of CP1,so we expect that the KMS indexes can be improved by imposing discrete symmetries.Because locallySO(4)～SU(2)1×SU(2)2,the parity group is

Apart from this there is anothersymmetry which swaps the two SU(2)'s whose generator is

Multiplying the elements inZ2×Z2by τ,we can generate more elements:

Averaging over the full discrete groupwe end up with the final KMS indexes

Indeed the spin-4 quantum conserved charge is recovered.

For the higher rank case,the lettersstill transform in the bi-fundamental representation ofSO(N)1×SO(N)2therefore the character is also given by a product of two individual characters:

We find that all the higher-spin KMS indexes are negative.Imposing the parity groupZ2×Z2will not help.For examples,one can obtain

WhenN＞ 4,the subgroups are not Abelian and the representation R is not reducible so that we do not have thesymmetry anymore.Therefore,we conclude that KMS index fails to predict the existences of the quantum conserved higher-spin currents10Here we have not considered the Pfaffian currents which could give a spin-N conserved currents [12].for the coset models SO(2N)/SO(N) ×SO(N) whenN＞ 2.

Note that in [14],it was found with the brutal force method that the cosets SU(N)/SO(N) and SO(2N)/SO(N) ×SO(N) possess the spin-4 quantum conserved currents.They used similar letters as ours in the counting.The crucial difference is that their letters jMare defined in the whole algebra while ourskαonly have the coset components.By lifting the letters with a conjugation11Basically,j M ～ gk αg -1 with g ∈G.into the whole algebra they can construct the gauge invariant operators from the trace operators.As we explained in section 3,this counting is incomplete.

4.3.CPN coupled with fermions

In this section,we generalize the KMS index to include fermionic letters.We have seen that the CPNmodels are not quantum integrable.However it has been known for a while that the quantum integrability of the CPNmodels can be restored by adding massless Dirac fermions[15].To illustrate our construction,we focus on this model but our method is generally applicable.

Without imposing the charge conjugation at the beginning,the KMS index can be computed in the presence of the fermionic letters.The fermions are chiral so the possible letters are

which give rise to the character

This character is problematic because in the conformal block the conformal dimension takes half-integer value such that the inversion formula does not work anymore.To cure this we can consider the ‘bosonization’ of the model by gauging the symmetryU(1) ×U(1) .For this gauge group we introduce two more auxiliary parameters and modify the fermionic character as

Recall the bosonic character is

Combining these two letters we can define the total partition function as a productZ=ZB ZF

If we integrate out the auxiliary parametersz i,yiwe end up with the generating function without half-integer conformal block contributions because integrating out the gauge symmetryU(1) ×U(1) guarantees the fermionic letters to group in pairs.

As argued the generating function will not contain unwanted characters corresponding to half-integer conformal dimensions.Note that we can not use the exponential form of the partition function to do this integral directly because it is not well-defined due to the appearance of the square root in the exponent.Instead we should understand it as an expansion form so we introduce another parameter with respect to which we can do the expansion

If we want to compute the index up to J = 6,the expansion up to the power u8is enough.The resulted KMS indexes are

Thus the GW argument fails.Now let us impose the charge conjugation symmetry.In other words,we need consider the charge conjugation invariant letters.The bosonic part can be treated in the same way.Gauging the Z2charge conjugation symmetry means that we should consider the real fermionic letters

Even though these letters are bosonic,we need to take into account of the Pauli’s exclusive principle

Therefore the partition function can be computed as

where the infinite products can be expressed with the qpochhammer symbols.Using this fermionic partition function,one can find that all the indexes are zero.The vanishing of the KMS index is due to the chiral structure.Therefore,the final indexes (3.19) are simply given by

The negative KMS indexes show that the GW argument fails again.

In the literature,CPNmodels are not often expressed as a coset model.Instead they are expressed in terms of complex vectors.We can also compute the KMS index in this formalism.The action is a complex version of (2.5):

The single-letters are

and the corresponding character is given by

If we want to impose the charge conjugation symmetry,the real single-letters are

with the character

Combining with the ferminonic parts(4.33)we reproduce the exactly the same KMS indexes (4.34) for smallJ＜ 7.

5.Summary

In this note,we elaborated the Komatsu,Mahajan and Shao’s index of quantum integrability which systematized the analysis of Goldschmidt and Witten’s argument.As applications,we revisited some quantum integrable coset modelsand found the following results:

1.The algebraic structure of the letters is crucial,particularly when it is trivial the KMS index vanishes for coset models.

2.The KMS indexes of the O(N) model in the coset description depend on N whenN＜ 7.WhenN≥ 7 the KMS indexes will be stable.After imposing the discrete symmetry,the KMS indexes become independent of N and predict the existences of spin-4 and spin-6 conserved currents.

3.After imposing the discrete symmetries,the coset modelhas KMS index I (4) =1 suggesting the existence of a spin-4 conserved currents.

4.The indexes of the coset modelswhenN≥3 andare all non-positive.The results are in conflict with the ones[14].The reason is that in[14]the letters used in the counting are defined in the whole group G instead of the coset G/H.It implies the operators which are built from these letters are not only invariant under H-transformation but also under Gtransformation.Therefore only a subset of the gauge invariant operators can be constructed from the letters used in [14] so the counting there is incomplete.

We also extended the KMS analysis to the theories with fermions and studied the CPNmodel coupled with massless Dirac fermion.We found that KMS index in this kind of model failed to predict any high-spin conserved currents.Our analysis suggests that in order to have positive KMS index one has to consider coupling the fermions with non-trivial algebraic structure.For example,it would be interesting to consider the KMS index in the supersymmetric theories[16].

Acknowledgments

JT would like to thank Shota Komatsu for his inspiring lectures on integrability at the 13th Kavli Asian Winter School.The work was in part supported by NSFC Grant No. 11335012,No. 11325522 and No.11 735 001.

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