FFIAN/TD-01/21IITP/TH-01/21ITEP/TH-01/21MIPT/TH-01/21 to the memory ofSergey Natanzon
Generalized Q-functions for GKM
A. D. Mironov a,b,c ∗ , A. Morozov d,b,c † a Lebedev Physics Institute, Moscow 119991, Russia b ITEP, Moscow 117218, Russia c Institute for Information Transmission Problems, Moscow 127994, Russia d MIPT, Dolgoprudny, 141701, Russia
Abstract
Recently we explained that the classical Q Schur functions stand behind various well known propertiesof the cubic Kontsevich model, and the next step is to ask what happens in this approach to the generalizedKontsevich model (GKM) with monomial potential X n +1 . We suggest to use the Hall-Littlewood polynomi-als at the parameter equal to the n -th root of unity as a generalization of the Q Schur functions from n = 2to arbitrary n >
2. They are associated with n -strict Young diagrams and are independent of time-variables p kn with numbers divisible by n . These are exactly the properties possessed by the generalized Kontsevichmodel (GKM), thus its partition function can be expanded in such functions Q ( n ) . However, the coefficientsof this expansion remain to be properly identified. At this moment, we have not found any “superintegra-bility” property < character > ∼ character , which expressed these coefficients through the values of Q atdelta-loci in the n = 2 case. This is not a big surprise, because for n > Q functions are notlooking associated with characters. Serezha Natanzon was a very original scientist. He taught us a lot about his beloved Hurwitz numbers andrelated world of interesting special functions. One of these lessons was about the Q Schur functions, which, in hisopinion, had to be applied to counting of holomorphic coverings with spin structures (spin Hurwitz numbers),and they really did [1]. Once one knows about the Q Schur functions, it gets immediately clear that they areapplicable to the Kontsevich model [2], because they have just the needed properties: do not depend on eventimes, are labeled by special Young diagrams, satisfy the BKP equations, and behave nicely under action ofthe Virasoro algebra [3]. Since the Q Schur functions are related to characters (of the Sergeev group [4, 5]), itcomes without a surprise that they exactly what is necessary to formulate the superintegrability property [6, 7] < character > ∼ character of the Kontsevich model [8, 9], moreover, as usual in this case, the coefficients atthe r.h.s are made from the same Q Schur functions at special delta-loci. At least in this case, this follows frommagnificent factorization formulas for the Q Schur characters [10, 11].Knowing all this, one is tempted to move towards the generalized Kontsevich model [12, 13] (GKM), witha monomial potential X n +1 to begin with. The first obvious question is what are the relevant Q ( n ) functions,which were just Schur’s Q = Q (2) for n = 2. In this paper, we suggest as a plausible candidate the Hall-Littlewood polynomials at the n -th root of unity, which is a straightforward generalization of one of the manydefinitions at n = 2. We explain that at least some of the needed properties are captured by this suggestion.But some are not, and Serezha is no longer here to teach us how to resolve these problems. They remain forthe future work. ∗ [email protected]; [email protected] † [email protected] a r X i v : . [ h e p - t h ] J a n Generalized Q Schur polynomials
In this section, we briefly list the main properties of a generalization of Q Schur polynomials from n = 2 to n >
2. To this end, we need a couple of new quantities: for the Young diagram ∆ = { δ ≥ δ ≥ . . . δ l ∆ > } = { m , m , . . . } , we define the standard symmetry factor z ∆ := ∞ (cid:89) a =1 m a ! · a m a (1)and two less conventional n -dependent quantities L ∆ := ∞ (cid:89) a =1 m a (cid:89) k =1 β k (2)and β ∆ := l ∆ (cid:89) i β δ i (3)where β ( n ) k := 1 − ω kn = 1 − e πikn (4)Thus L ∆ is vanishing when diagram ∆ has n (or more) lines of the same length. Roots of unity of the degreedictated by power of the monomial potential in the GKM is a new twist in the QFT-Galois relation withfar-going applications.The properties of our new functions Q ( n ) R { p } are: • They are equal to the special values of the Hall-Littlewood polynomials [14]: at t = ω n which is theprimary n -th root of unity Q ( n ) R := (cid:112) L R · Mac R ( q = 0 , t = e iπ/n ) (5)where Mac R denotes the Macdonald polynomial. Because of the factor L R , Q ( n ) R is non-zero only for theYoung diagram R which has no more than n − S n P , and its subset of diagrams of size | R | = m , by S n P ( m ). • Hereafter we deal with the symmetric functions Q ( n ) R of variables x i as (graded) polynomials of powersums p k := (cid:80) i x ki , which we call time-variables (they are proportional to times of integrable hierarchyassociated with the GKM). The polynomials Q ( n ) R { p } are independent of the time-variables p kn . Thisproperty is preserved by an arbitrary rescaling of time-variables, but we do not apply this rescaling, anduse the choice associated with the Cauchy formula in the form (15) below. • They form a closed algebra Q ( n ) R { p } · Q ( n ) R { p } = (cid:88) R ∈ R ⊗ R R ∈ SnP N RR ,R Q ( n ) R { p } (6)i.e. the Littlewood-Richardson coefficients of Macdonald polynomials vanish when q = 0 , t = ω n and R , R ∈ S n P and R / ∈ S n P . Like in the case of n = 2, Mac R ( q = 0 , t = ω n ) themselves do not vanish for R / ∈ S n P , and then they can also depend on even p nk , thus the set of Q ( n ) R { p } is a sub-set of that of theHall-Littlewood polynomials, and it is a non-trivial fact that it is a sub-ring . • The Fr¨obenius formula for the generalized Q Schur polynomials is Q ( n ) R { p k } = (cid:88) ∆ ∈ O n P Ψ ( n ) R (∆) z ∆ p ∆ (7)2t the l.h.s. R ∈ S n P . The set O n P at the r.h.s. consists of all diagrams with line lengths not divisibleby n , and this formula reflects a remarkable one-to-one correspondence between O n P and S n P : these setshave the same sizes, and the map is non-degenerate as follows from the orthogonality relations: (cid:88) ∆ ∈ O n P Ψ ( n ) R (∆)Ψ ( n ) R (cid:48) (∆) β ∆ z ∆ = δ RR (cid:48) , (cid:88) R ∈ S n P Ψ ( n ) R (∆)Ψ ( n ) R (∆ (cid:48) ) = β ∆ z ∆ δ ∆∆ (cid:48) (8)Hence, one can construct an inverse map p ∆ = (cid:88) R ∈ S n P Ψ ( n ) R (∆) Q ( n ) R { p k } β ∆ (9) • In the scalar product (cid:68) p k (cid:12)(cid:12)(cid:12) p l (cid:69) = kβ k · δ k,l (10)the Q -functions are orthogonal: (cid:68) Q ( n ) R (cid:12)(cid:12)(cid:12) Q ( n ) R (cid:48) (cid:69) = || Q ( n ) R || · δ R,R (cid:48) (11)with || Q ( n ) R || = 1 (12) • As usual, one can introduce the skew Q -Schur functions Q ( n ) R/P defined as Q ( n ) R { p + p (cid:48) } = (cid:88) P ∈ SP Q ( n ) R/P { p } Q ( n ) P { p (cid:48) } (13)They are given by Q ( n ) R/P { p } = (cid:88) P ∈ SP N RP S Q ( n ) S { p } (14) • Q Schur polynomials satisfy the Cauchy formula, (cid:88) R ∈ S n P Q ( n ) R { p } · Q ( n ) R { Tr X k } = exp (cid:32)(cid:88) k =1 β ( n ) k p k Tr X k k (cid:33) (15)Since β ( n ) k vanishes whenever k is divisible by n , the r.h.s. is independent of all p kn . We write thesecond set of times in Miwa variables, p (cid:48) k = Tr X k , because we need this form of the Cauchy formula inconsideration of correlators below. • The Virasoro and W algebras act rather simple on the generalized Q Schur polynomials, see sec.4. • The Q polynomials themselves are not τ -functions of the KP hierarchy and its reductions (like the KdVand Boussinesq ones). Instead, for n = 2 they satisfy the BKP hierarchy [15, 16, 1] which does not yethave any direct counterpart for n > • The main difficulty at this stage is that there is yet no formula for Q ( n ) R per se, without referring to theHall-Littlewood and Macdonald polynomials. Indeed, the ordinary Schur polynomials at n = 1 have adeterminant representation [14], or can be realized as an average over charged fermions (see a review in[17]); the Q Schur polynomials at n = 2 have a Pfaffian representation instead of the determinant one(see, e.g., [1, Eq.(74)]), or can be realized as an average over neutral fermions [18, 17, 15, 16, 11]; whathappens for n > n = 2 case.3 Monomial GKM
The monomial Generalized Kontsevich model is defined by the N × N Hermitian matrix integral [12] Z ( n ) ( L ) := N ( L ) · (cid:90) exp (cid:18) − Tr X n +1 n + 1 + Tr L n X (cid:19) dX (16) Z ( n ) ( L ) depends only on the eigenvalues of the background matrix field L , and, with a proper choice of thenormalization factor N ( L ), it can be treated as a formal series either in positive or in negative powers of L [19]. In fact, Z ( n ) ( L ) is a symmetric functions of the eigenvalues λ ± i of the external matrix L , and, hence, canbe considered as a function of the power sums or the “time-variables” p ± k := tr L ± k . These two cases requireproper (different) choices of the normalization factors and are referred to as character and Kontsevich phases[19]. In this paper, we are interested in the more sophisticated Kontsevich phase, and in what follows we omitthe superscript ”-”: p k := p − k .The potential in the exponent has an extremum at X = L , and, in the Kontsevich phase, one expandsaround it in inverse power of L . In this phase, one has to choose the normalization factor N ( L ) := exp (cid:18) n + 1 Tr L n +1 (cid:19) exp (cid:32) − (cid:88) a + b = n − Tr L a XL b X (cid:33) dX (17)This provides that Z ( n ) ( L ), which depends on the eigenvalues λ i of the matrix L , can be understood as a formalpower series in λ − i , and, in fact, is a power series in p k := Tr L − k [12].It possesses more advanced definitions as a D -module and/or peculiarly reduced KP τ -function. Namely, • For a given n the partition function Z ( n ) ( L ) is actually independent of p kn , this explains the choice ofnotation for the potential: what matters is usually not the potential X n +1 but its derivative X n . • Z ( n ) ( L ) as a (symmetric) function of λ i is a τ -function of the KP hierarchy in Miwa variables, i.e. satisfiesthe bilinear difference Hirota equations and can be expressed as a determinant. Z ( n ) ( L ) as a functionof power sums p k /k is a τ -function of the KP hierarchy in the ordinary higher time variables (hence,the name “time-variables” for p k ), and satisfies the bilinear differential Hirota equations. Moreover, fora given n , it is actually an n - reduction of the KP hierarchy, say, the KdV hierarchy for n = 2, or theBoussinesq hierarchy for n = 3. • Z ( n ) ( L ) satisfies the Ward identities [20, 21]. When rewritten in terms of p − k , these constraints form Borelsubalgebras of the Virasoro and W -algebras, ˆ W ( p ) m Z ( L ) = 0 with 2 ≤ p ≤ n , m ≥ − p [12, 22]. In thecharacter phase, the Ward identities in terms of p + k are rather the ˜ W -constraints [19]. • The lowest of these constraints, ˆ L − Z ( n ) ( L ) = ˆ W (2) − Z ( n ) ( L ) = 0 called string equation along with theintegrable hierarchy equations generates the whole set of the Ward identities.We see that this list has some parallels with the list of properties of the Q -functions in the previous section.Thus it comes without a surprise that • Z ( n ) ( L ) in the Kontsevich phase can be expanded in functions Q ( n ) { p } .This character expansion is the subject of the present paper. We will see that it is not yet as powerful as inthe case of n = 2 [9], still generalization to n > One can calculate the GKM integral (16) perturbatively. To this end, one has to expand around the extremumof potential at X = L , i.e. to shift X = L + Y , and deal with the integral Z ( n ) ( L ) == (cid:90) exp (cid:32) − Tr ( L + Y ) n +1 − L n +1 n + 1 + L n Y + 12 (cid:88) a + b = n − Tr L a Y L b Y (cid:33) exp (cid:32) − (cid:88) a + b = n − Tr L a Y L b Y (cid:33) dY (18)4xpanding the first exponential and evaluating the obtained Gaussian integral. The measure in this integral isdefined so that < > = 1.Thus, we define the correlation function by the Gaussian integral (cid:104) . . . (cid:105) := (cid:90) . . . exp (cid:32) − (cid:88) a + b = n − Tr L a Y L b Y (cid:33) dY (19)and first evaluate the propagator. In terms of the eigenvalues λ i of L , the propagator is (cid:104) Y ij Y kl (cid:105) n = δ il δ jk (cid:88) a + b = n − λ ai λ bj (20)When this does not lead to a confusion, in what follows we omit the index n in the notation of the average, butwe should remember that the propagator depends on n and has grading level, i.e. the power in L − equal to n − Correlation functions with the propagator (20) have complicated denominators and often can not be expressedin terms of the time-variables p k = tr L − k = N (cid:88) i =1 λ − ki (21)From this perspective it looks like a miracle that there are many exceptions: plenty of admissible correlatorsexist, i.e. those expressible through the time-variables. In particular, as we already pointed out an importantresult from the theory of GKM [12] is that Z ( n ) ( L ) in the Kontsevich phase actually is a power series in timevariables.Thus we understand that at least the correlators which comes from perturbative expansion of the GKM areadmissible. In fact, expanding exponential in (18), one obtains rather sophisticated averages (cid:42)(cid:32) Tr ( L + Y ) n +1 − L n +1 n + 1 − L n Y − (cid:88) a + b = n − Tr L a Y L b Y (cid:33) m (cid:43) n (22)and they should depend on λ i only through p k . For the ordinary Kontsevich model with n + 1 = 3, thesecorrelators are just (cid:68)(cid:0) Tr Y (cid:1) m (cid:69) , but already in the quartic case, n + 1 = 4 they contain L : powers of Tr Y should be combined with those of Tr LY .These correlators do not exhaust all the admissible correlators, but, as a first step, we concentrate on thisspecial set in this paper. Now we are going to study the perturbative expansion of (18) as a function of time-variables. A natural fullbasis in the set of such functions is provided by characters , for instance, by the Schur functions χ R { p } , whichform a set labeled by the Young diagrams with a natural grading by the size of these diagrams | R | : Z ( n ) ( L ) = (cid:88) R C R χ R { p } (23)The question is what are the coefficients C R . Since Z ( n ) ( L ) is a KP τ -function, they satisfy the Pl¨ucker relations.But actually from the theory of GKM [12] we know more: for a given n , it is independent of all p kn and is a τ -functions of the (appropriately reduced) KP hierarchy. For example, at n + 1 = 3, it is a τ -function of theKdV hierarchy, which depends only on odd time-variables p k +1 . This means that χ R { p } is actually not themost adequate basis, because this type of reduction looks complicated in it. This is clear already from the caseof n = 2, where the coefficients C R are quite involved [23, 24].5rom what we already know from sec.2, it is clear that much better for a given n is a basis formed by the Q -functions Q ( n ) R with R ∈ S n P . Thus, more precisely, our interest is in Z ( n ) ( L ) = (cid:88) R ∈ S n P C ( n ) R Q ( n ) R { p } (24)As we demonstrated in [9], the coefficients C (2) R in this basis are very simple and natural in the case of n = 2,in contrast with expansion into C R . Remark.
It could look appealing to extract Q ( n ) at a special delta-locus, Q ( n ) R { δ k,n +1 } from the coefficients C ( n ) R : Z ( n ) ( L ) ? = (cid:88) R ∈ S n P c ( n ) R · Q ( n ) R { δ k,n +1 } Q ( n ) R { p } (25)like we did in [9] for n = 2. This may seem natural because applying the Cauchy identity to the original integral(16), one can conclude that e − n +1 Tr X n +1 = exp (cid:32) − (cid:88) k k Tr X k · δ k,n +1 (cid:33) = (cid:88) R ( − | R | Q ( n ) R ∨ { Tr X k } · Q ( n ) R { δ k,n +1 } (26)Of course, this is far from a reliable argument, and it is not a big surprise that things are not so simple. As wewill see shortly, the expansion (25) is actually not possible for n > n = 2 For n + 1 = 3 the relevant averages involve only L -independent operators: (cid:80) ∞ m =0 1(2 m )! · m (cid:10) (Tr Y ) m (cid:11) . Forexample, for the first two terms12! · (cid:10) (Tr Y ) (cid:11) = 148 · (cid:0) p + 4 p (cid:1) = 148 · (cid:32) Q (2)[2 , { p k } − √ Q (2)[3] { p k } (cid:33) == 132 · (cid:16) Q (2)[3] { δ k, } Q (2)[3] { p k } − Q (2)[2 , { δ k, } Q (2)[2 , { p k } (cid:17) (27)and 14! · (cid:10) (Tr Y ) (cid:11) = 19 · · (cid:0) p p + 25 p + 200 p p + 16 p (cid:1) == − · · (cid:32) √ Q (2)[6] { p k } − Q (2)[5 , { p k } + 7 Q (2)[4 , { p k } + (cid:112) [2] Q (2)[1 , , { p k } (cid:33) = (28)= 51024 · (cid:16) Q (2)[6] { δ k, } Q (2)[6] { p k } − Q (2)[5 , { δ k, } Q (2)[5 , { p k } − Q (2)[4 , { δ k, } Q (2)[4 , { p k } − Q (2)[1 , , { δ k, } Q (2)[1 , , { p k } (cid:17) We can note that the coefficient in front of Q (2)[1 , , { δ k, } Q (2)[1 , , is exactly the product of those in front of Q (2)[3] { δ k, } Q (2)[3] { p k } and Q (2)[2 , { δ k, } Q (2)[2 , { p k } , i.e. − = · (cid:0) − (cid:1) . This is a manifestation of the generalproperty [10] that the coefficient in front of Q (2) R { δ k, } Q (2) R { p k } in the character expansion of the cubic Kontsevichpartition function Z (2) ( L ) is factorized: equal tocoeff Q (2) R { δ k, } Q (2) R { p k } (cid:16) Z (2) ( L ) (cid:17) = l R (cid:89) i =1 f (2) ( R i ) (29)It is actually equal to [9]coeff Q (2) R { δ k, } Q (2) R { p k } (cid:16) Z (2) ( L ) (cid:17) = 12 | R | / · Q (2)2 R { δ k, } Q (2) R { δ k, } Q (2) R { δ k, } Q (2)2 R { δ k, } (30)which has exactly such factorization property due to elegant factorization identities, see [11] for details. Here2 R means the Young diagram obtained from R by doubling its line lengths.Thus, one finally obtains Z (2) ( L ) = (cid:88) R ∈ S P | R | / · Q R { δ k, } Q R { δ k, } · Q R { p k } Q R { δ k, } (31)6 .6 The basic example: n = 3 After reminding the already known situation at n = 2, we now make the first step into terra incognita at n > n + 1 = 4 (cid:10) Tr Y (cid:11) = (cid:104) Y ij Y jk Y kl Y li (cid:105) = 2 P ij,jk P kl,li + P ij,kl P jk,li == 2 δ ik ( λ i + λ i λ j + λ j )( λ k + λ k λ l + λ l ) + δ i,j,k,l ( λ i + λ i λ j + λ j )( λ j + λ j λ k + λ k ) == (cid:88) ijl λ i + λ i λ j + λ j )( λ i + λ i λ l + λ l ) + (cid:88) i λ i (32)This correlator is not expressed through the time variables. However, there is another term of the same grading ,i.e. of the same degree in L − , with two extra powers of L in the operator compensated by those in theextra propagator: (cid:28)(cid:16) Tr LY (cid:17) (cid:29) . When it is added to (cid:10) Tr Y (cid:11) with an appropriate coefficient, the sum getsexpressed through the time variables: − (cid:18)(cid:68) Tr Y (cid:69) + 2 (cid:28)(cid:16) Tr LY (cid:17) (cid:29)(cid:19) = p + 6 p p
36 = (33)= 136 (cid:16) β − / (cid:16) Q (3)[4] − √ Q (3)[2 , , (cid:17) − (2 √ i ) (cid:16) Q (3)[2 , − iQ (3)[3 , (cid:17)(cid:17) = β (cid:88) R ∈ S P | R | =4 c R · Q (3) R { δ k, } Q (3) R { p k } with c [4] = 7 , c [2 , = c [3 , = 1 − i √ , c [2 , , = − , , , S P , is indeed missing at the r.h.s.Similarly, in the next order12! · (cid:18)(cid:28)(cid:16) Tr Y (cid:17) (cid:29) − (cid:28)(cid:16) Tr LY (cid:17) · Tr Y (cid:29) + 43 (cid:28)(cid:16) Tr LY (cid:17) (cid:29)(cid:19) == 132 · · (cid:0) p p + 96 p p + 13 p + 156 p p p − p + 36 p p (cid:1) (35) = 132 · (cid:18) · · √ β Q (3)[8] + 5 · i √ Q (3)[7 , − ·
72 (3 √ i + 13) Q (3)[6 , − β (cid:112) β Q (3)[6 , , + 5 ·
72 (11 − √ i ) Q (3)[5 , + 3 · (cid:112) β Q (3)[5 , , ++ 5 · · √ β Q (3)[4 , + 5 √ β √ i + 11 √ (cid:16) Q (3)[4 , , − iQ (3)[4 , , (cid:17) − · √ β Q (3)[4 , , , − √ β (cid:112) β Q (3)[3 , , − · β (cid:16) Q (3)[3 , , , + iQ (3)[3 , , , (cid:17)(cid:19) = = (cid:18) β (cid:19) (cid:88) R ∈ S P | R | =8 c R · Q (3) R { δ k, } Q (3) R { p k } (36)with c [8] = c [4 , = 5 · · , c [7 , = c [6 , = c [5 , = − · · (1 + 4 i √ , c [6 , , = c [5 , , = c [3 , , = − ,c [4 , , = c [4 , , = − − i √ , c [4 , , , = − , c [3 , , , = c [3 , , , = 13 (37)At n = 3, one should not expect relations like (29) already because they do not respect the selection rule forpartitions from S P : say, [2 , ∈ S P , but [2 , , , / ∈ S P . Still, one can observe some interesting relations,which resemble the corollaries of (29): c [3 , , , = c [3 , , , = | c [2 , | = | c [3 , | c [4 , , , = c [4] · c [2 , , c [4 , , c [4] c [3 , = c [4 , , c [4] c [2 , (38)however, say, c [4 , (cid:54) = c (39)7e see from the above formulas that extracting Q (3) { δ k, } from the coefficients simplify them a little, butthe remaining pieces do not have any nice enough properties, e.g. do not factorize in the spirit of [10, 11], as theydid for n = 2. Worse than that, already in the next order, some of Q (3) R { δ k, } = 0, though the correspondingcontribution from the diagram R is non-vanishing: hence such an extraction is simply impossible in generalsituation. The first diagrams with this property for n = 3 appear at the level twelve: [5 , , , , , , ,
1] and[6 , , , , · (cid:18)(cid:28)(cid:16) Tr Y (cid:17) (cid:29) − (cid:28)(cid:16) Tr LY (cid:17) · (cid:16) Tr Y (cid:17) (cid:29) + 4 (cid:28)(cid:16) Tr LY (cid:17) · Tr Y (cid:29) − (cid:28)(cid:16) Tr LY (cid:17) (cid:29)(cid:19) == − p p
324 + 5 p p
324 + 7 p p
162 + 25 p p p
972 + 5 p p p − p p
162 + 25 p p p − p p p
81 + p p p
162 + 325 p −− p p p p p − p p p p p p p (cid:18) β (cid:19) (cid:88) R ∈ S P | R | =12 C R · Q (3) R { p k } (40)Wherever possible, we present the much simpler and more “symmetric” expressions for c R defined from C R = c R · Q (3) R { δ k, } c [12] = c [8 , = 5 · · · ,c [11 , = c [10 , = c [9 , = − · · · (59 + 54 i √ , c [7 , = c [6 , = 5(2423 − · i √ ,c [10 , , = c [9 , , = c [3 , , , , , = 175 , c [8 , , = c [8 , , = 175(3 i √ − , c [7 , , = 35(15 i √ − ,c [7 , , = − i √ , c [6 , , = − − i √ , c [6 , , = − · − i √ c [6 , , = − · i √ , c [5 , , = 25 · − i √ , c [5 , , = − ·
23 + 132 i √ ,c [8 , , , = c [4 , , , , = − , c [7 , , , = c [7 , , , = 35(23 − i √ , c [6 , , , = c [5 , , , = 5(333 − i √ ,c [6 , , , = 5(179 − i √ , c [5 , , , = c [5 , , , = 5(113 − i √ , c [4 , , , = c [4 , , , = 5 323 − i √ ,c [4 , , , = − , c [4 , , , , = c [4 , , , , = 5(17 + 18 i √
3) (41)but in the above mentioned cases, when Q (3) R { δ k, } = 0 for R = [5 , , , , [5 , , , , , [6 , , , , C R ’smake sense: C [5 , , , = − · i √ C [5 , , , , = 3 / √ · · √ i ) − − i )128 C [6 , , , , = 3 / √ · · i ) + 19 √ − i )128 (42) The generators of the positive part ( m >
0) of Virasoro algebra areˆ L ( n ) m := (cid:88) k =1 ( k + nm ) p k ∂∂p k + nm + 12 nm − (cid:88) k =1 k ( nm − k ) ∂ ∂p k ∂p nm − k (43)It acts on the linear space of Schur functions, moreover, it leaves the sub-space S n P intact, so thatˆ L ( n ) m Q ( n ) R { p } = (cid:88) R (cid:48) , | R (cid:48) | = | R |− mn ξ ( n,m ) R,R (cid:48) Q ( n ) R (cid:48) { p } (44)e.g. ˆ L ( n ) m Q ( n )[ r ] = rQ ( n )[ r − mn ] (45)8 .1 n = 2 For n = 2 its action is known on Q (2) with time-variables, rescaled by √ L (2) m Q (2) R (cid:26) p k √ (cid:27) = l R (cid:88) i =1 ( − ) ν i ( R i − m )( √ δ Ri,m · Q (2) R − m(cid:15) i (cid:26) p k √ (cid:27) (46)where R − m(cid:15) i means that exactly i -th length is diminished: R i −→ R i − m . This can make it shorterthan some other lines and thus imply reordering of lines in the diagram to put them back into decreasingorder, then ν i ( R, m ) is the number of lines, which the i -th one needs to jump over, e.g. ˆ L (2)2 Q (2)[6 , , (cid:110) p k √ (cid:111) =(6 − Q (2)[5 , , (cid:110) p k √ (cid:111) − (5 − Q (2)[6 , , (cid:110) p k √ (cid:111) and ˆ L (2)3 Q (2)[7 , , (cid:110) p k √ (cid:111) = (7 − Q (2)[6 , , (cid:110) p k √ (cid:111) − (6 − √ Q (2)[7 , (cid:110) p k √ (cid:111) . If R i − m = 0, then the line is simply omitted and the coefficient 1 / √ Z (2) GKM in the basis Q (2) R (cid:110) p k √ (cid:111) israther ugly. Expansion is nice in terms of Q (2) per se, instead the Virasoro action on Q (2) per se is slightlymore involved than (46): ˆ L (2) m Q (2)[ r ] = rQ (2)[ r − m ] ˆ L (2) m Q (2)[ r, = rQ (2)[ r − m, + √ Q (2)[ r +1 − m ] ˆ L (2) m Q (2)[ r, = rQ (2)[ r − m, + 2 Q (2)[ r +1 − m, , m ≥ . . . (47) n For generic n , we note that the Virasoro algebra acts in the simplest way to slightly renormalized functions Q ( n ) R = β l R / Q ( n ) R . Now the action of the operator ˆ L ( n )1 on Q ( n ) R is expanded into the Q Schur functions at level | R | − n . The rule is that ˆ L ( n )1 Q ( n ) R spans only by Young diagrams ˇ R = R − k i (cid:15) i − k j (cid:15) j , k i + k j = n , and one of k i can be zero. This means that, for instance, in the case of n = 3, there can be only either diagrams R − (cid:15) i ,or R − (cid:15) i − (cid:15) j . Suppose that ˇ R do not requite re-ordering the lines (i.e. the decreasing order is still preserved),and, moreover, the lines in all diagrams have different lengths (i.e. they are strict partitions). Then,ˆ L (3)1 Q (3) R = (cid:88) i R i Q (3) R − (cid:15) i + β (3)2 (cid:88) i>j Q (3) R − (cid:15) i − (cid:15) j + β (3)1 (cid:88) i>j Q (3) R − (cid:15) i − (cid:15) j (48)Similarly, in the case of n = 4, there are possibilities: R − (cid:15) i , R − (cid:15) i − (cid:15) j and R − (cid:15) i − (cid:15) j so thatˆ L (4)1 Q (4) R = (cid:88) i R i Q (4) R − (cid:15) i + β (4)3 (cid:88) i>j Q (4) R − (cid:15) i − (cid:15) j + β (4)2 (cid:88) i>j Q (4) R − (cid:15) i − (cid:15) j + β (4)1 (cid:88) i>j Q (4) R − (cid:15) i − (cid:15) j (49)and generally ˆ L ( n )1 Q ( n ) R = (cid:88) i R i Q ( n ) R − n(cid:15) i + n − (cid:88) k =1 β ( n ) k (cid:88) i>j Q (4) R − k(cid:15) i − ( n − k ) (cid:15) j (50)Moreover, this action is immediately continued to action of the general ˆ L ( n ) m . For instance, instead of (48), onehas now ˆ L (3) m Q (3) R = (cid:88) i R i Q (3) R − m(cid:15) i + β (3)2 (cid:88) i>j Q (3) R − m(cid:15) i − m(cid:15) j + β (3)1 (cid:88) i
0, the Virasoro action is not injective, thus (45) and (60) are not enough to checkany constraint, one needs bigger pieces of matrices ξ and ζ . For n = 2, the constraints with m ≥ − n >
2, one needs to add similar W -constraints up to W ( n ) , which can be studied in a similar way:ˆ W ( p | n ) m · (cid:88) R c ( n ) R Q ( n ) R { p } = 0 , m ≥ − p, p = 2 , . . . , n (62)Here W ( p | n ) denotes the W algebra of spin p : p = 2 corresponds to the Virasoro algebra, etc.For instance, in the case of n = 3, one has to add to the Virasoro constraints (57) the W (3 | -algebraconstraints. This algebra at generic n looks like [25, 12, 22]:ˆ W (3 | n ) m := (cid:88) k =1 ( k + l + nm ) P k P l ∂∂p k + l + nm + (cid:88) k =1 nm + k − (cid:88) l =1 l ( nm + k − P k ∂ ∂p l ∂p nm + k − l ++ 13 mn − (cid:88) k =1 mn − k − (cid:88) l =1 kl ( mn − k − l ) ∂ ∂p l ∂p k ∂p mn − k − l + 13 − mn − (cid:88) k =1 − mn − k − (cid:88) l =1 P k P l P − mn − k − l (63)where P k := p k − nδ n +1 ,k at k >
0, and P k = 0 otherwise. To conclude, we described an interesting set of functions Q ( n ) , which, in many respects, generalize the Q Schurfunctions Q (2) , and provide a promising basis to expand the partition function of GKM in the Kontsevich phase: (cid:90) exp (cid:18) Tr X n +1 n + 1 + Tr L N X (cid:19) dX ∼ (cid:88) R ∈ S n P C ( n ) R · Q R { Tr L − k } (64)This basis is distinguished by the selection rules: independence of p kn and of diagrams beyond S n P , and,perhaps, also by integrability and Virasoro/W properties of Q ( n ) , which still need to be carefully formulated.However, unlike the n = 2 case, the coefficients C ( n ) R are not properly identified and interpreted, largelybecause of the calculation difficulties with Q ( n ) functions. In the n = 2 case, these coefficients have a verynice form [9] C (2) R = Q R { δ k, } Q R { δ k, } Q R { δ k, } which has a profound combinatorial explanation [10, 11], and can berelated to the superintegrability property of the cubic Kontsevich model. At the moment, it is an open problemif so defined superintegrability persists for GKM at n >
2. We discuss this question and a related issue ofclassification of correlators in the GKM elsewhere.
Acknowledgements
We are indebted to Sasha Alexandrov, John Harnad and Sasha Orlov for numerous comments on [9], whichlargely stimulated our further work in this direction.This work was supported by the Russian Science Foundation (Grant No.20-12-00195).11 ppendix
In this Appendix, we illustrate our consideration by first terms of the Q -expansion of the GKM partitionfunctions in the n = 4 case.The partition function of quintic GKM is defined as Z (4) = (cid:28) exp (cid:18) −
15 Tr Y − Tr LY − Tr (cid:16) L Y + LY LY (cid:19)(cid:29) := (cid:68) exp (cid:16) − V − V | − V | (cid:17)(cid:69) (65)We introduced here a convenient notation V a | b for a term of the form Tr L a X b . Then the grading level, i.e. thepower of an average (cid:81) m V a m | b m in L − , is equal to (cid:80) m (cid:0) b m − a m (cid:1) .The lowest grade terms in expansion of the partition function are: Grading 5. (cid:28) − V | + 12 V | (cid:29) = (cid:28) − Tr LY + 12 (cid:16) Tr (cid:16) L Y + LY LY (cid:17)(cid:17) (cid:29) == 132 (cid:113) β (4)1 Q (4)[5] − − i ) Q (4)[4 , − i ) Q (4)[3 , − (1 + 4 i ) √ Q (4)[3 , , + (4 − i ) √ Q (4)[2 , , + β (4)3 √ − iQ (4)[2 , , , == p + 4 p p + 4 p p
32 = 5 β (4)3 (cid:88) R ∈ S P | R | =5 c R · Q (4) R { δ k, } Q (4) R { p k } c [5] = 9 , c [4 , = c [3 , = 3(1 − i ) , c [3 , , = c [2 , , = − (1 + 4 i ) , c [2 , , , = − Grading 10. (cid:28) V V | + 12 V | − V | V | + 124 V | (cid:29) = (cid:28)
15 Tr Y · Tr (cid:16) L Y + LY LY (cid:17) + 12 (cid:0) Tr LY (cid:1) −−
12 Tr LY (cid:16) Tr (cid:16) L Y + LY LY (cid:17)(cid:17) + 124 (cid:16) Tr (cid:16) L Y + LY LY (cid:17)(cid:17) (cid:29) = (cid:32) β (4)3 (cid:33) (cid:88) R ∈ S P | R | =10 c R · Q (4) R { δ k, } Q (4) R { p k } c [10] = c [5 , = 3 · , c [9 , = c [8 , = c [7 , = c [6 , = − i ) ,c [8 , , = c [7 , , = c [6 , , = c [6 , , = − i ) , c [5 , , = c [5 , , = − i ) , c [4 , , = c [4 , , = 3(11 − i )5 ,c [7 , , , = c [6 , , , = − , c [5 , , , = c [5 , , , = − − i )5 , c [4 , , , = − − i ) ,c [4 , , , = − − i ) , c [4 , , , = c [3 , , , = − − i )5 , c [3 , , , = 1 ,c [3 , , , , = 3(7 − i ) , c [4 , , , , = c [4 , , , , = 3(7 + 6 i ) , c [3 , , , , = 39 , c [5 , , , , = − Similarly to the n = 3 case, Q (4) R { δ k, } = 0 for R = [3 , , , , , C R makes sense for this diagram: C [3 , , , , , = − √ β (4)3 (67)Note that R = [3 , , , , ,
1] is the only Young diagram out of S P (10) that contains 6 lines. Grading 15.
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