Non-Abelian W-representation for GKM

$W$-representation is a miraculous possibility to define a non-perturbative (exact) partition function as an exponential action of somehow integrated Ward identities on unity. It is well known for numerous eigenvalue matrix models when the relevant operators are of a kind of $W$-operators: for the Hermitian matrix model with the Virasoro constraints, it is a $W_3$-like operator, and so on. We extend this statement to the monomial generalized Kontsevich models (GKM), where the new feature is the appearance of an ordered P-exponential for the set of non-commuting operators of different gradings.


Introduction. Hermitian model and the idea of W -representation
Partition function of matrix models [1] usually satisfies an exhaustive set of Virasoro and W -constraints, which are, however, not so easy to solve. For example, for the Hermitian matrix model with the partition function Z N {p} where N is the size of matrix, the Ward identities read [2] L n Z N {p} = 0, n ≥ −1 (1) and the operatorŝ L n := k (k + n)p k ∂ ∂p k+n + n−1 a=1 a(n − a) ∂ 2 ∂p a ∂p n−a + 2N n ∂ ∂p n + N 2 δ n,0 + N p 1 δ n+1,0 − (n + 2) ∂ ∂p n+2 (2) form a Borel subalgebra of the Virasoro algebra. The underlined term breaks the grading, the grading of p k being k. Such a choice of this term corresponds to choice of the Gaussian phase. In this phase, this system of equations has a unique solution [3,4], which is given by [5] (see [6][7][8] for early precursors) wherê As explained in [9] (see also [10]), this representation can be deduced from the fact that the Virasoro constraints (2) can be all encoded in a single equation that has a unique solution. The operators commute in the simple way: l 0 ,Ô 2 = 2Ô 2 (6) andl 0 · 1 = 0 (7) i.e.l 0 is the grading operator, and the grading ofÔ 2 is 2. This is the main point: we combined Virasoro constraints in such a way that the grading-breaking piece is converted into grading operatorl 0 . Now introducing the grading parameter x via the rescaling p k → x k p k , one comes to the equation with an obvious solution Since the solution is unique, one establishes that (9) provides a representation of the Hermitian matrix model partition function. With our operators we do not need x, and (5) just has (3) as an obvious solution.
Since [5], there were many more examples of W -representations for many different models [12], see [9] for a recent summary and for an evidence for unambiguity of solutions. However, there remains an important exception from the general list: the monomial Kontsevich models [13,14] beyond the simplest cubic example [11]. The goal of this letter is to fill the gap and provide a simple description of what happens to W -representation for the generalized Kontsevich model (GKM).
The partition function of the monomial GKM is given by the matrix integral over N × N Hermitian matrix X, and depends on the external matrix M . At large M (so called Kontsevich phase [15]), this partition function is understood as a power series in time-variables p k := Tr M −k , the coefficients of this power series being independent of the size of matrix N . Hence, the notation Z r (M ) = Z r {p}. This partition function does not depend on p rk -variables, and is normalized so that lim M→∞ Z r (M ) = Z r {0} = 1. The Ward identities of this matrix model are described by constraints from the W (r) -algebra and become rather involved at large r. In the next section, we consider the simplest case of r = 2, when they form a Borel subalgebra of the Virasoro algebra. In section 4, we consider the first non-trivial case of r = 3, when the W -algebra constraints emerge for the first time, and, in section 5, we consider the generic r case.

W-representation in cubic case
We start with the partition function (10) with r = 2 [16]. In this case, the partition function satisfies the Virasoro constraints [17][18][19], Here the sums over k and a run over odd numbers. These constraints can be encoded in a single equation that has a unique solution, This equation contains the terms of gradings 0 and 3. The zero grading term comes from the last (underlined) term in (12) and isl so that (13) takes the form with the operators of grading 3 beinĝ The commutation relation is Introducing the grading parameter x via the rescaling p k → x k p k , we come to the equation Its solution is exponential, which is nothing but the standard W -representation [9,11].

W-representation in quartic case
Now we consider the r = 3 case. This is the first truly non-trivial case. We have now a combination of Virasoro and W -constraints [20][21][22][23] (3) where P k := p k − 3 · δ k,4 , and a, b, c, k, l in the sums are not divisible by 3. They can be combined into a single equation that unambiguously determines the partition function where the parameter c can be chosen rather arbitrarily (only non-negative rational c can give rise to additional superfluous solutions of this equation) [9]. At the l.h.s. of this equation, there are operators of gradings 0, 4 and 8.
For the special choice of c = −1, the coefficients in front of the sum n (3n − 1)p 3n−1 ∂ ∂p3n−1 , coming from the first term in (22), and in front of the sum n (3n − 2)p 3n−2 ∂ ∂p3n−2 , coming from the second term, are equal to each other, so that the zero grading operator is nothing butl 0 with k not divisible by 3. With this choice, (22) looks like where the operators of gradings 4 and 8 arê and the sums over k, l, a, b, c run over positive integers not divisible by 3.
The commutation relations are Introducing the grading parameter x via the rescaling p k → x k p k , we come to the equation Its solution is going to be an ordered exponential with the operatorsÔ 4,8 depending on α andÔ 4 , on the constant c. Another possibility is to define yet another operator of zero grading, and deal with the equation It makes the whole consideration more involved. In particular, at some peculiar rational values of c, there is a degeneration, which gives rise to additional superfluous solutions to Eq. (22). For instance, at c = +2, one gets and the coefficient α is not determined from Eq. (22).
The simplest way to generate this expansion is as follows. Let us look for a solution in the form Z 4 {p} = k x 4kΨ k · 1. Then, (29) is equivalent to the recursion relation with the initial conditionsΨ 0 = 1,Ψ 1 =Ô 4 .
Note that the recursion relation is consistent with similar relations obtained by J.Zhou [24], though we derive them within a different framework. However, the operatorsÔ 4 andÔ 8 do not commute and, hence, do not lead to a simple exponential W -representation form of (30) (this is not quite consistent with the conclusion of [24]): The series (30)   Let us briefly sketch the next r = 4 case. This time we should use the following constraints: The corresponding W algebra can be expressed in terms of bosonic currents: where (n) r denotes n modulo r. The currents are: and the normal ordering implies all the derivatives moved to the right. The sums in these expressions run over integers not divisible by 4. The last term in 4Ŵ As usual [9] we consider a peculiar linear combination of these constraints: and according to [9] this equation has a unique solution for almost arbitrary constants c i . For our current purposes they can be chosen so that the zero grading operators combine into l 0 = p k ∂ ∂p k . This choice is It deserves making a brief remark on grading. If we neglect the shift of the fifth time in (37), then all the terms coming from 4Ŵ (4) n−4 have the grading 15. The third term in (36) contains at maximum one shift, which means there is also a term of grading 10. The second term contains terms with one or two shifts, which means there are terms of grading 5 and 10. Hence, the zero grading terms come only from the leading term in (36) and, similarly, in (34) and (35). This immediately gives (39).
For this choice (39) the equation for the partition function acquires the form Then the W -representation is given by: We illustrate this representation by evaluation of the partition function Z 4 {p} in Appendix B up to order 15.
In the 5-th and 10-th order, available at [26], it coincides with the answer in that paper.

Towards arbitrary r
Partition function Z r in the GKM (10) with potential x r+1 does not depend on p nr and satisfies the whole set of W -constraints of the orders ranging from 2 (Virasoro) to r [13], and the W -generators are defined in [25] and in Appendix B, the first two being They can be expressed through the Z r -twisted scalar fields [25], and higher constraints are rather involved, e.g.
and rŴ (5) : J n1 J n2 J n3 : − 1 6r 4 n1+n2+n3=rn n 1 r + n 2 r + n 3 r : J n1 J n2 J n3 : In these formulas, and the sums run over integers not divisible by r. We denote n r := (n) r r − (n) r where, as before, (n) r is the value of n modulo r. At the moment, we introduce a parameter µ in the term violating the grading in order to control the grading easier. We will ultimately put µ = 1.
The leading term of the W -generator is These W -constraints can be combined into a single equation again with the nearly arbitrary constants c i . This equation is a sum of operators of gradings {(r + 1)i}, i = 0..r − 1, which is given by the expansion of the operators into operators of definite gradings: rŴ n,j has grading rn − j(r + 1).
Again the constants c i can be adjusted so that all the zero grading operators, i.e. all the terms in n=1 p nr−i ∂ ∂p nr−i with all i = 1, . . . , r, come with the unit coefficients and combine into the grading operator This is the choice c i = (−1) i . Then, introducing the grading parameter x via the rescaling p k → x k p k , we come to the equation As we explained, the term with j = 0 in this sum reproduces the operatorl 0 , and we finally come to the equation (we put µ = 1 here) The solution to this equation is the iterated integral withÂ(t) := r k=2 kt kÔ (r+1)k , t := x r+1 , i.e. the series where some i k can be the same. The coefficient is the repeated integral In the commuting case, the coefficients would sum up just to

To W -representation from matrix Ward identity for GKM
An interesting option would be to start directly from the identity [13,17,21,27] for the matrix integral from which we extract the GKM partition function Z V depending only on negative powers of the matrix variable M , p k = tr M −k . For the monomial potentials V r (X) = X r+1 r+1 , this means that M r = L, and Z V = Z r turns out to be independent of all p rn , see [13,14] for details. In this case, and substitution into (58) gives a sum of terms with r + 1 different gradings, associated with r derivatives of the exponential. If we multiply the equation by M and take a trace, the gradings (powers of M −1 ) will be n(r + 1) with n = −1, 0, . . . , (r − 1). Actually the lowest grading with n = −1 does not show up, because The most interesting is grading 0, where we get the operatorl 0 . Indeed, There are two other contributions in this grading, which do not contain p-derivatives of Z r : one appears when the L derivative acts on det V ′′ (Λ) instead of Z r , another one, when two L derivatives act twice on the same exponential. Analysis in other gradings gets more involved and will be addressed elsewhere.

Conclusion
In this letter, we resolve a puzzle of the W -representation [5] for the monomial generalized Kontsevich models [13] beyond the cubic case [11]. As usual, the deviation from the standard situation appeared very simple but unexpected and implies far-going consequences. It turned out that the W -representation is not an ordinary exponential but an ordered P -exponential of a linear combination of non-commuting W -like operators of different gradings. We remind that, like many other matrix models [1], the GKM partition function is a KP τ -function [13], thus what we observe is a striking appearance of P -exponential in the field of integrable systems. This brings the seemingly simple matrix models into a direct contact with Yang-Mills theories, where the P -exponentials play the central role: as predicted long ago, the non-Abelian nature has no conflict with integrability.
In the narrower field of matrix models per se, the W -representations provide a truly effective method for generating as many terms of the GKM partition function as one needs. This opens new possibilities for study of these very interesting and archetypical models. Some details are still lacking, and we have not yet derived a truly closed expression for arbitrary r, this is one of the simplest subjects for the future work.

(k) n
In this Appendix, we describe how one can obtain the relevant W -operators rŴn (k) by the normal ordering of a product of currents [25]. The main point is that the spectral curve for the monomial GKM model Z r (which can be obtained from the corresponding loop equations [19]) is the r-sheeted covering of a sphere, which is clear both from the integrable hierarchy point of view (since the system is described by the r-th reduction of the KP hierarchy) [13,25], and from the topological recursion point of view [3]. This is why it is natural to define the current to be J(z) := J n z −n/r−1 with the current modes given by (47). This expression involves the r-th root of z, and, hence, one has to specify which of the roots is used (the sheet of the covering). We denote choosing the m-th root as z m . One can arbitrarily choose the ordering of z m , m = 1, . . . , r, but, for the sake of definiteness, we choose them to be z 1/r m+1 = exp 2πi r · z 1/r m and denote z 1 = z. Note that integer powers of z are the same for all z m , but, at the level of J(z), the arguments are all different, and one can use non-singular operator expansions. At the same time, the final answer contains only integer powers of z.
Now the procedure of constructing the rŴn (k)-operators consists of three steps.

The starting point is an auxiliary operator
which is very simple and general, but not normally ordered.

2.
One has to normally order rŴ (k) aux (z) in such a way that all positive current modes are moved to the right.
3. After normal ordering, one has to omit all the current modes divisible by r: J nr = 0 in order to finally obtain rŴ (k) (z). W (2) n : In this case, we have

Now note that both rŴ
where we denoted ω m := exp 2πim r and used that [J n1 , J −n2 ] = n 1 δ n1,n2 , and, hence, the anomaly term is Calculating the sum over n requires a regularization as usual for the anomaly. At last, in order to rewrite the remaining sum in the last line of (66), we use the identity so that we finally obtain The terms linear in J in the second line are omitted at the third step, since they are proportional to J nr . This is evident since the whole expression should be single-valued, and, hence, it depends only on integer powers of z, i.e. on J nr . It can be manifestly seen in the following way: it is a sum of three terms of the form 1 r 3 1≤m1<m2<m3≤r n1,n2>0,n3 with two other terms corresponding to [J n1 , J −n3 ]J n2 and [J n3 , J −n2 ]J n1 . The sum of these three terms is proportional to giving rise to the sum The last line in (70) is due to the identity for any n 1 , n 2 , n 3 / ∈ rZ (73) n : In this case, the calculation is very similar, but there is a subtlety. That is, one needs a counterpart of formulas (68) and (73). Now it, however, has a more subtle structure: the r.h.s. depends not only on r and on divisibility of the sum n 1 + n 2 + n 3 + n 4 by r, but also on divisibility of pairs n 1 + n 2 , etc. Indeed, the identity is : J n1 J n2 J n3 J n4 : In order to calculate the terms of the form : JJ :, one proceeds similarly to (67) and uses the identity (ω m1 − ω m2 ) 2 = r 2 · r 2 − 1 12 − r 2 · (n 1 ) 2 r + (n 2 ) 2 r − 1 δ n1+n2,rn for any n 1 , n 2 / ∈ rZ (76) Here the symmetrization symbol Sym ωm i means that we sum over all permutations of ω i . At last, in order to evaluate the remaining constant anomaly term, one twice uses the summation as in (67), and the identity 1≤m1<m2<m3<m4≤r Sym ωm i ω m1 ω m2 (ω m1 − ω m2 ) 2 ω m3 ω m4 (ω m3 − ω m4 ) 2 = 8r · (r 2 − 1)(r − 2)(r − 3)(5r + 7) 5760 (77) in order to ultimately obtain (45).