A new kind of anomaly: on W-constraints for GKM

We look for the origins of the single equation, which is a peculiar combination of W-constrains, which provides the non-abelian W-representation for generalized Kontsevich model (GKM), i.e. is enough to fix the partition function unambiguously. Namely we compare it with the scalar projection of the matrix Ward identity. It turns out that, though similar, the two equations do not coincide, moreover, the latter one is non-polynomial in time-variables. This discrepancy disappears for the cubic model if partition function is reduced to depend on odd times (belong to KdV sub-hierarchy of KP), but in general such reduction is not enough. We consider the failure of such direct interpretation of the “single equation” as a new kind of anomaly, which should be explained and eliminated in the future analysis of GKM.


Introduction
This is going to be a rather technical paper, targeted at clarification of the long-standing puzzles of Generalized Kontsevich model (GKM) [1]- [15]. It is not fully successful, still it can attract attention to potentially important aspects of the story. No doubt, at technical level these observations are well known to people who worked with GKM, but we make an attempt to summarize them and promote to a more conceptual level. This is needed because of the new accents introduced into the GKM theory quite recently, in [16][17][18][19][20][21][22] and [23]- [25], and the need to explain the origins and the form of the "single equation" [24] and the character expansion in terms of Hall-Littlewood polynomials [16][17][18][19][20][21][22]. We do not achieve these goals in the present paper, but we try to better explain the difficulties of one particularly promising suggestion from [25] -with the hope that it gains attention and will be somehow resolved in the near future.

Single Equation (SE) :
While rather well established in the case of the ordinary cubic (r = 2) Kontsevich model [1], these issues are quite difficult to address for r > 2. It is the purpose of this paper to make one more technical step in this direction.
Namely, we study a scalar implication of matrix Gross-Newmann equation (1.2), and the suggestion of [25] to use it as SE -a basic equation which provides Z r as a unique solution in the form of the non-Abelian W -representation. In this paper we call it "the main equation", or just ME to simplify the reference. In other words, the main question of the present paper is if and, if not, what is the difference.

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Substitution of a badly controlled system of matrix equations (1.2), which is believed to be equivalent to an infinite set of W -constraints (1.3), by a single equation SE [24] is a big simplification -surprisingly this is possible without a loss of information. However, to make it fully satisfactory, we need a maximally simple origin if this SE -and ME would be just a dream. Unfortunately, as anticipated in [25], the story is not just so simpleand details, though seemingly technical, are quite interesting. The fact that the simplest Ward identity ME is not quite the same as SE, which controls the solution, is an interesting twist of the story and this is what we call anomaly in the title of this paper: The actual calculation consists of three steps. First one needs to express matrix derivatives through eigenvalues, this is discussed in a separate section 3. Then one needs to act on the product Z r [L] = Z cl r [M ] · Z r {p} and convert the equation w.r.t. eigenvalues into the one for the "quantum" Z r {p}, depending on time variables. And afterwards one should interpret the results. We demonstrate that "anomaly" has two origins. The first is that the coefficients of the terms with derivatives over p kr are ugly and, actually, non-polynomial in time variables. This can serve as a possible interpretation of the need for the r-reduction, i.e. the need for these derivatives to vanish -what looks particularly convincing in the case of cubic (r = 2) model, when this is the only manifestation of the anomaly. Unfortunately, for r > 2 the situation gets more obscure. The second phenomenon is that for r > 2 this non-polynomiality shows up also in the coefficients of other derivatives -and ME is not sufficient to explain the vanishing of these unwanted contributions. Of course, other constituents of (1.3) should imply this nullification, but this brings us back to the complicated form of (1.3) and SE (1.4).

Strong dependence on N : the classical piece of partition function
Usually in discussion of GKM we emphasize the remarkable property 3) from above list -that the essential ("quantum") part of partition function depends on the matrix size N only through the choice of the locus p k = tr M −k -an N -dimensional non-linear subspace in the infinite-dimensional space of time-variables {p k }. The shape of Z{p k } is, however, independent of N , and in this sense the N -dependence of Z{p} on N is weak.
In this paper we switch the accent to another side of the story: to the "classical" part of partition function, which is much simpler, but depends on N much stronger -and this will have a serious impact on the Ward identities (1.3) and (1.4), making their simplest treatment through the otherwise appealing "main equation" (1.5) less straightforwardif not totally meaningless.
For monomial potential V (M ) = M r+1 r+1 the classical part of partition function can also be easily expressed through the time variables p k = tr M −k , though expressions are a little lengthy. They are naturally written through the Schur functions S R {p}, where R denotes the Young diagrams (for example, S [1] = p 1 , S [2]  important, these formulas have strong and explicit dependence on N : As we will see, non-trivial Schur functions in the pre-exponent survive in the main equation (1.5) and make it non-polynomial in time variables. The only case when this does not matter at all, is r = 1, which we will briefly mention in section 4 below. In conventional cubic model at r = 2 the non-polynomiality can be eliminated by r-reduction (from KP to KdV in this case) -this we will see in section 5. Starting from r = 3, however, the problem (anomaly) is far more difficult to cure, and the corrected form of ME -and thus the simple derivation of SE -still needs to be found.

JHEP10(2021)213 3 From matrices to eigenvalues
As already mentioned, GKM (1.1) is an eigenvalue model, the integral is reduced to eigenvalues of X and the answer depends on the eigenvalues of L = M r . Still the reason for the special properties of GKM is that originally it depends on the matrix variable, and the natural Ward identities [45][46][47][48][49][50]) are matrix-valued -given by (1.2). Since they contain matrix derivatives, it is separate exercise to convert them to the eigenvalue form. What we need are diagonal elements of ∂ r Z[L] . Still the answer is well known from perturbation theory in quantum mechanics [51] (where one diagonalizes the Hamiltonian and obtains corrections to the wave functions): according to [2], Note that λ js can coincide, also with λ i -then one should apply the l'Hopitale rule, and this gives rise to more sophisticated structures. In [52] a special technique was developed on this occasion. We, however, just work with explicit formulas, without going into details of the derivations. In particular, one can explicitly check, that in accordance with this general prescription.

JHEP10(2021)213 4 A toy example at r = 1
This is a special case, where equation (1.2) has a "wrong" power of ∂/∂L. It was used as a training example in [2]. For and superscript label (1) refers to r = 1. This is the ordinary Virasoro operator, which defines Virasoro constraints in Hermitian matrix model, and it appears here because this model can be also treated as GKM with additional insertion of a power of det X in the integral, what causes also an increase of r by one [7][8][9][10][11][12]53].

Implication from the known Z 2
We now proceed to the study of the true main equation (1.5), beginning from the first case of cubic Kontsevich model. What we need is to substitute

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First of all we can substitute the known series for Z 2 {p k } (last time cited in the appendix to [25]), and we expect to get zero. It is instructive to see how this really works. If we substitute instead of Z 2 just 1 -the first term in the series,-we get a polynomial of grading 3: , then a polynomial of grading 6 and so on: the more gradings we include into Z 2 {p}, the higher is the grading of (5.2): if gradings up to 3m are included into Z 2 {p}, then (5.2) is of grading 3(m + 1). Thus we obtain zero for (5.2) in the sense that every particular grading vanishes as we include appropriately many terms into Z 2 {p}. Also at every stage the answer is not just of definite grading, it is actually a polynomial in time variables.
One can wonder, what happens to S [1] factor in (2.2) -why does not it produce nonpolynomial contributions? It turns out to be a rather delicate adjustment. Already the quadratic singularity S −2 The linear singularity is even more miraculous: potentially relevant terms in (2.2) are and the term 1 µ 1 +µ 2 is independent of α, but depends in β. It vanishes when γ = 3β − 2, what includes the true values β = γ = 1, but clearly demonstrates the delicate balance between various contributions. Therefore it is not surprising that thus balance will be often violated -the surprise is that it continues to hold (anomaly is lacking) for r = 2 for arbitrary N , and also for the coefficients of odd derivatives ∂Z ∂p 2n+1 -as we will explain in the next sections.

r = 2, all times
The next exercise is to convert (5.2) into an equation for Z 2 {p k }, similar to what we considered above in section 4. Assume first that Z 2 {p k } depends on all the time-variables p k . Then, once again substituting (5.1) into (5.2) we get: The last term is of course the same as we got from substitution of 1 into (5.2). However, with the first derivatives of Z there is a trouble (underlined): for derivatives w.r.t. even p 2k the coefficients are not polynomial in p. Moreover, they depend on N , with somewhat sophisticated self-consistency/reduction relations between different N . In terms of Schur This problem of non-polynomiality is cured (the underlined terms are absent) in the action on functions Z 2 {p 1 , p 3 , p 5 , . . .}, which depend only on odd times p 2k−1 .

r = 2, odd times = cubic Kontsevich model
Assume now that Z{p k } depends on all the odd time-variables p 2k−1 . Then the terms with ξ (N ) drop out of (5.5) and we get a differential equation for Z −2 {p} with polynomial JHEP10(2021)213 coefficients in p: and the grading-counting operator Note that elimination of derivatives ∂Z ∂p 2k automatically eliminates all even times p 2k from the coefficients of (5.7): for r = 2 the r-reduction is necessary and sufficient for ME to reproduce SE. 6 The first non-standard case: quartic model r + 1 = 4

Solution to projected Ward identity
Now we can repeat all the same steps in the first non-trivial case of quartic GKM with r = 3. We will see that the non-polynomiality gets now even more pronounced.
In terms of µ-variables (λ i = µ r i ) the main equation in this case looks as follows: For r > 3 denominators get larger and degenerations provide higher derivatives of Z.

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The next step is to substitute and obtain an equation ME (1.5) for Z 3 with time-derivatives instead of the µ-ones.
Then we can compare it to SE (1.4), which in the case of r = 3 involves two operators [25]: 4 and a, b, c,  . This will be one of the apparent differences from the ME, which contains third derivatives of all the four kinds.

Non-trivial denominators
The other striking difference will be non-polynomiality. In fact, one can observe it at the very early stage. For r > 2 it is enough to look at the derivative-free term in ME. Namely, if Z 3 = 1 the l.h.s. of (6.1) is non-vanishing, but contains contributions of just two (r − 1) gradings: −4 and 0. In the simplest case of N = 2 Z = 1 (6.1)
While the first polynomial piece is stabilized and does not vary anymore for N > r, the shape of non-polynomial terms is not stable -it varies with N . Building up the true Z 3 {p} results into the shift of the two non-vanishing pieces to infinite gradings p ∞ , p ∞ -and in this sense the answer, understood as the contributions at every particular grading, gets vanishing.
The moral is that now the non-polynomiality is less related to r-reduction: a function can be independent of p rk (like Z 3 = 1), still (6.1) does not convert it into a polynomialnon-trivial denominators occur. However, the proper Z 3 {p} is converted to zero. Together with occurrence of the underlined term in (6.4) this implies that at least some terms in the main equation with derivatives of Z 3 {p k } w.r.t. p k should be non-polynomial, even if k is not divisible by r. Since such non-polynomiality does not appear in the highest-derivative terms ∂ r Zr ∂p i 1 ...∂p ir , the natural guess after that is that these additional terms are made from the lower W -constraints, i.e. from the complements of the main equation (1.5) -the other corollaries of the matrix Ward identity (1.2).

ME for r = 3, all times
As we already know from the previous subsection, there will be problems with relating SE to ME. In addition to the two nice terms at the r.h.s. of 0 (1.5) n 1 n 2 n 3 p n 1 +n 2 +n 3 +8 ∂ 3 Z ∂p n 1 ∂p n 2 ∂p n 3 + + 1 3 ∞ n 1 ,n 2 n 1 n 2 p n 1 +n 2 +4 ∂ 2 Z ∂p n 1 ∂p n 2 + . . . (6.6)

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the non-polynomial terms will appear, which depend on N . In the simplest case of N = 2 the full expression is: n1n2n3pn 1 +n 2 +n 3 +8 27 [4,4] + 4pn 1 +n 2 −2S [5,5] 18 + Like it was for r = 2, in (6.6) there are items with the derivatives ∂/∂p 3k , which are absent in (6.7). They can be eliminated by asking Z 3 to belong to the 3-reductionexactly like it happened in the previous section for r = 2. This is the positive part of the story: cancellation of anomaly requires the r-reduction. But is the r-reduction sufficient for deriving SE from ME?
Unfortunately, the answer is "no": now there are a few more striking differences between (6.6) and (6.7), e.g.
i) already in (6.6) there are items with p 3k in the sums, which are absent in (6.7),
These are the qualitative deviations, as to the quantitative details of the two formulas, they look even more different. Still both are true. The only way out of this apparent discrepancy is that the anomalous difference between the two formulas is made from some other W -constraints (1.3), not incorporated into the simple equation (6.7). This would mean that literally SE = ME even for r-reduced Z r , still the anomaly is canceled by r-reduction plus some additional information -superficial for the scalar projection ME of the Ward identity (1.2), still implied by the entire (1.2). This is indeed the case, but it is quite difficult to see. We show how it works for contributions from a few lowest gradings 4m to Z 4 = ∞ m=0 z 4m . In grading four we have exact matching: both (6.6) and (6.7) while (for N = 2) ME : −np n ∂z 8 ∂p n + p 4 + 6p 2 p 2 1 9 z 4 − 4(7S [10] + 7S [9,1] + 10S [8,2] ) 27S [2] z 0 + 2] ) ∂z 4 ∂p n + + 1 3 n 1 ,n 2 n 1 n 2 p n 1 +n 2 +4 ∂ 2 z 4 ∂p n 1 ∂p n 2 = 0 (6.10) Already at this level the difference between the two correct formulas looks quite pronounced -and it only increases at the next levels. Some new ideas are needed to express the (vanishing) difference in terms of the W-constraints (1.3) and, hopefully, find a concise and universal expression for this discrepancy. Since it relates two clearly distinguished quantities -the SE which is a single polynomial equation, which defines Z{p k }, and ME which is the distinguished scalar projection of the fundamental matrix Ward-identity (1.2)

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-there should be some simple relation between them. We see that the hope of [25], that this relation is just an identity, fails. But in the simplest cases (like the basic Kontsevich model r = 2) it is true -and thus the discrepancy is an anomaly, in the sense which still remains to be formulated. Anyhow, so far anomalies were always comprehensiblehopefully this will be the case with this new one as well.

Conclusion
In this paper we studied the properties of the main equation (1.5) from [25]. This is important because this equation seems to somehow accumulate the power of all the Wconstraints in monomial GKM and fully define the time dependence of its partition function Z r {p}. In particular it should imply that this partition function is independent of p rk (of time-variables with the numbers divisible by r). We demonstrated that it does so in an elegant way: if there was a p rk -dependence in Z r , we would not get an equation for it, which is polynomial in time-variables p k . Since Z r is known from [2,4,5] to be a KP τ -function (this is relatively simple to demonstrate), independence of p rk means that it belongs to the r-reduction of KP hierarchy. In fact, one can consider our calculation as a new kind of a proof of this statement (that Z r is an r-reduced τ -function), but still a rather sophisticated and undirect one. A concise, clear and direct proof remains highly desirable. Also desirable is a direct relation of our calculation with the elegant description [27,28] of the W -constraints for r-reductions as a normal ordering of "circular formula" r m=1 J z · e 2πim/r . There are now few doubts that the W -constraints are implied by the single main equation -but the way it works remains unclear. One can only hope that if this is clarified, the constraints will also come in some clever form -probably, provided by the circular formula.
Our main result is that the main equation (1.5), directly following from the matrix Ward identity (1.2), is not exactly the same as the "single equation" (SE) of [24,25], but differs from it by additional non-polynomial (!) terms, which presumably are proportional to The fact that some other W constraints emerge in addition to the single equation in the truly-first-principle approach (based on [26]) can be important for better understanding of its surprising predictive power -a possibility for a single equation to substitute the JHEP10(2021)213 entire set of the W -constraints (which has more than one generator: already two, L −1 and L 2 , for r = 2). As a byproduct of our calculation we found an amusing structure of non-polynomial terms, with a peculiar embedded dependence on N . Since non-polynomial terms are coefficients of ∂Z r /∂p kr which actually vanish for the GKM partition function, the true significance of these formulas, at least in the case (b), remain unclear -still they look interesting by themselves and can show up in some other contexts. The observation of "anomaly" SE = ME even for r-reduced τ -functions leaves the puzzle of W -constraints and the origin of W -representation for GKM with r > 2 [25] unsolved. This adds to the equally puzzling complication of superintegrability formulas and character calculus for r > 2: at least the appropriate basis of Q-functions [16][17][18][19][20][21][22] remains unknown. It is unclear if there is a direct connection between these two complications -anyhow, the story of GKM is still incomplete and at least one additional idea is still lacking. Of course the previous ideas, like "circular formula" [27,28] and non-abelian Wrepresentation [24,25], also need to be polished and brought to the same level of clarity as determinantal representation and KP integrability [4,5], -but this is hard to do before the "anomaly" issue is fixed, which controls the puzzle of r-reduction and the very origin of sophisticated W -constraints and the way they follow from the apparent original Ward identity [26]. If superintegrability and character expansion will also get clarified by the resolution of this puzzle, or need to wait for additional insights, remains to be seen.
Last but not the least -all the formulas in this paper are obtained for particular low values of r and N , what is enough to reveal the emerging structures and phenomena. Still general consideration and proofs remain to be given. They can also bring new ideas and further develop and clarify the theory of GKM, which remains mysteriously complicated and transcendent -perhaps a little less now, but still far from simplicity and transparency achieved for the other eigenvalue matrix models (including the cubic GKM).