Fermionic approach to weighted Hurwitz numbers and topological recursion A. Alexandrov1,2,4∗ , G. Chapuy5† , B. Eynard2,6‡ and J. Harnad2,3§ 1 Center
for Geometry and Physics, Institute for Basic Science (IBS), Pohang 37673, Korea 2 Centre
de recherches math´ematiques, Universit´e de Montr´eal C. P. 6128, succ. centre ville, Montr´eal, QC H3C 3J7 Canada 3 Department
of Mathematics and Statistics, Concordia University 1455 de Maisonneuve Blvd. W. Montreal, QC H3G 1M8 Canada
4 ITEP,
Bolshaya Cheremushkinskaya 25, 117218 Moscow, Russia 5 CNRS
UMR 7089, Universit´e Paris Diderot Paris 7, Case 7014 75205 Paris Cedex 13 France 6 Institut
de Physique Th´eorique, CEA, IPhT, F-91191 Gif-sur-Yvette, France CNRS URA 2306, F-91191 Gif-sur-Yvette, France
Abstract A fermionic representation is given for all the quantities entering in the generating function approach to weighted Hurwitz numbers and topological recursion. This includes: KP and 2D Toda τ -functions of hypergeometric type, which serve as generating functions for weighted single and double Hurwitz numbers; the Baker function, which is expanded in an adapted basis obtained by applying the same dressing transformation to all vacuum basis elements; the multipair correlators and the multicurrent correlators. Multiplicative recursion relations and a linear differential system are deduced for the adapted bases and their duals, and a Christoffel-Darboux type formula is derived for the pair correlator. The quantum and classical spectral curves linking this theory with the topological recursion program are derived, as well as the generalized cut and join equations. The results are detailed for four special cases: the simple single and double Hurwitz numbers, the weakly monotone case, corresponding to signed enumeration of coverings, the strongly monotone case, corresponding to Belyi curves and the simplest version of quantum weighted Hurwitz numbers. ∗
e-mail: e-mail: ‡ e-mail: § e-mail: †
[email protected] [email protected] [email protected] [email protected]
1
Contents 1 Weighted Hurwitz numbers and generating τ -functions 1.1 Weighted Hurwitz numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 KP and 2D-Toda τ -functions of hypergeometric type as generating functions 1.3 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 2 5 6
2 Infinite Grassmannian, Baker functions, adapted bases 2.1 The formal infinite Grassmannian, τ -functions and Baker functions . . . . . 2.2 Adapted bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7 7 10
3 Fermionic representations 3.1 Fermionic Fock space . . . . . . . . . . . . . . . . . . . . . 3.2 Fermionic construction of τ -functions, Baker function and adapted bases . . . . . . . . . . . . . . . . . . . . . . . 3.3 Multiplicative recursion relations for general gˆ . . . . . . . . 3.4 Euler derivative relations for general gˆ . . . . . . . . . . . . 3.5 The n-pair correlation function and Christoffel-Darboux type 3.6 Multicurrent correlator for general gˆ . . . . . . . . . . . . .
. . . . . . . . .
12 12
. . . . . . . . . . . . kernel . . . .
. . . . .
15 18 20 21 24
4 Specialization to the case of hypergeometric τ -functions 4.1 Convolution products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Adapted basis, τ -function and Baker function for the hypergeometric case . . 4.3 An alternative realization of convolution actions . . . . . . . . . . . . . . . . 4.4 Recursion relations in the hypergeometric case . . . . . . . . . . . . . . . . . 4.5 The spectral curve and Kac-Schwarz operators . . . . . . . . . . . . . . . . 4.5.1 The quantum and classical spectral curve . . . . . . . . . . . . . . . . 4.5.2 Kac-Schwarz operators . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 The pair correlation function in the hypergeometric case . . . . . . . . . . . 4.7 Finiteness and generating function for Christoffel-Darboux matrix . . . . . . 4.8 The multicurrent correlators as generating functions for weighted Hurwitz numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25 25 26 28 31 33 33 34 36 37
5 Bose-Fermi correspondence and “cut and join” operators 5.1 Bose-Fermi correspondence and cut and join representation of the τ -function 5.2 Cut and join equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41 41 44
6 Examples 6.1 Simple (single and double) Hurwitz numbers [35, 39] . . . . . . . . . . . . . .
45 45
2
. . . . .
. . . . .
. . . . .
. . . . .
40
6.2 6.3 6.4
Three branch points (Belyi curves): strongly monotonic paths in the Cayley graph [9, 22, 29] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Signed Hurwitz numbers: weakly monotonic paths in the Cayley graph [16,20,22] 47 Simple quantum Hurwitz numbers [20, 22] . . . . . . . . . . . . . . . . . . . 47
Appendix A Fermionic representation of multipair correlators ˜ 2g (z, w) . . . . . . . . . . . . . . . . . . . . . . A.1 Fermionic representation of K g ˜ 2n A.2 The n-pair correlator K (z, w) . . . . . . . . . . . . . . . . . . . . . . . . . ˜ 2 (z, w) in the adapted basis . . . . . . . . . . . . . . . . . . . A.3 Expansion of K
1
49 49 50 52
Weighted Hurwitz numbers and generating τ -functions
It has been known since the work of Pandharipande and Okounkov [35,39] that certain special classes of τ -functions of the KP and 2D-Toda hierarchies can serve as generating functions for various types of weighted Hurwitz numbers. It also is known that the computation of these enumerative geometrical/combinatorial invariants may be embedded into the framework of Topological Recursion [10, 12, 13, 33]. Fermionic representations have played a key rˆole in simplifying the calculations and clarifying the origin of these constructions. The present work is part of an ongoing series (see [1, 2]) devoted to the inclusion of the generating function approach to weighted Hurwitz numbers of the most general type into the framework of Topological Recursion.
1.1
Weighted Hurwitz numbers
For a set of partitions {µ(i) }i=1,...,k of weight |µ(i) | = N , the Hurwitz numbers H(µ(1) , . . . , µ(k) ) are defined geometrically [25, 26] as the number of inequivalent N -fold branched coverings C → P1 of the Riemann sphere having k branch points with ramification profiles given by the partitions {µ(1) , . . . , µ(k) }, normalized by the inverse 1/| aut(C)| of the order of the automorphism group of the covering. An equivalent combinatorial/group theoretical definition [14,15,42] is that H(µ(1) , . . . , µ(k) ) is 1/N ! times the number of distinct factorizations of the identity element I ∈ SN in the symmetric group in N elements as a product of k factors hi , belonging to the respective conjugacy classes cyc(µ(i) ) hi ∈ cyc(µ(i)) ).
I = h1 , · · · hk ,
(1.1)
The equivalence of the two follows from the monodromy homomorphism from the fundamental group of P1 punctured at the branch points into SN obtained by lifting closed loops from the base to the covering. Denoting by `∗ (µ) := |µ| − `(µ) (1.2) 3
the colength of the partition µ (the difference between the weight and the length), the Riemann Hurwitz theorem relates the Euler characteristic χ of the covering curve C to the sum of the colengths `∗ (µ(i) ) of the ramification profiles at the branch points χ = 2 − 2g = 2N −
k X
`∗ (µ(i) ),
(1.3)
i=1
where g is the genus of C. ˜ A weight generating function G(z) and its dual G(z) := 1/G(−z), may be represented in terms of an infinite set of constants (or indeterminates) c := (c1 , c2 , . . . ),
(1.4)
either as an infinite product, or an infinite sum: ∞ ∞ Y X G(z) := (1 + zci ) = 1 + gi z i
˜ G(z) :=
i=1 ∞ Y
(1.5)
i=1 −1
(1 − zci )
=1+
i=1
∞ X
g˜i z i ,
(1.6)
i=1
where the coefficients {gi }i=1,...,∞ , {˜ gi }i=1,...,∞ in the (formal) Taylor series are, respectively, the elementary and complete symmetric functions gi = ei (c),
g˜i = hi (c).
(1.7)
We define the (associative, commutative) algebra K = Q[g1 , g2 , g3 , . . . ]
(1.8)
of polynomials in the gi ’s over the field Q of rationals, which is the same (when the number of ci ’s is finite) as the algebra Λ of symmetric functions of the ci ’s (over the coefficient field Q). Given a pair of partitions (µ, ν) of N , the weighted double and single Hurwitz numbers are defined by the sums [20, 22] HGd (µ, ν)
:=
d X
X0
k=0
µ(1) ,...,µ(k) Pk
i=1
HGd˜ (µ, ν)
:=
X0
k=0
µ(1) ,...,µ(k) i=1
(1.9)
WG˜ (µ(1) , . . . , µ(k) )H(µ(1) , . . . , µ(k) , µ, ν),
(1.10)
`∗ (µ(i) )=d
d X
Pk
WG (µ(1) , . . . , µ(k) )H(µ(1) , . . . , µ(k) , µ, ν)
`∗ (µ(i) )=d
HGd (µ) := HGd (µ, ν = (1)N ),
HGd˜ (µ) := HGd˜ (µ, ν = (1)N ) 4
(1.11)
X0 where denotes the sum over all k-tuples of partitions {µ(1) , . . . , µ(k) } other than the cycle type of the identity element (1N ), the weights WG (µ(1) , . . . , µ(k) ) := mλ (c) =
X 1 |aut(λ)| σ∈S
k
and
∗ (−1)` (λ) X WG˜ (µ , . . . , µ ) := fλ (c) = |aut(λ)| σ∈S
(1)
X
(1.12)
1 k cλiσ(1) , · · · cλiσ(k) ,
(1.13)
1≤i1 i. (2.32) This means that W ∈ GrH+ (H) must be in the “big cell”, but imposes no further restriction, since a representative of the left coset under the stabilizer of H+ may always be chosen in this lower triangular form. 12
It also implies that the sums (2.26) - (2.29) are semi-infinite, with the highest power of z in the Fourier series for wkg (z) and wkg∗ (z) being k − 1 wkg (z)
k−1 X
=
g−j−1,−k z j
(2.33)
j =−∞
wkg∗ (z)
k−1 X
=
−1 gk−1,j zj ,
(2.34)
j=−∞
and j
z =
j+1 X
−1 g−k,−j−1 wkg (z)
(2.35)
gj,k−1 wkg∗ (z).
(2.36)
k=−∞ j
z =
j+1 X k=−∞
3
Fermionic representations
3.1
Fermionic Fock space
We recall the conventions and notation for the fermionic operator approach to τ -functions developed by Sato and his school [27, 40, 41]. Using the notation of Section 2.1, we denote the semi-infinite wedge product space (Fermonic Fock space) F = Λ∞/2 H = ⊕N ∈Z FN
(3.1)
where FN denotes the charge-N sector, N ∈ Z. The pairs (λ, N ) consisting of a partition λ of any weight (with the convention that λi = 0 for i > `(λ)) and an integer N label the basis element in F. These are denoted |λ; N i := e`1 ∧ e`2 ∧ · · · ∈ FN ,
(3.2)
`i := λi − i + N
(3.3)
where is a strictly decreasing sequence of integers saturating, after a finite number of terms, in consecutively decreasing integers. These are viewed as the integer lattice points of occupied sites, and the vacuum element in each sector is given by the trivial partition |N i := |0; N i = eN −1 ∧ eN −2 ∧ · · · .
13
(3.4)
The Fermi creation and annihilation operators {ψi , ψi∗ }i∈Z are the generators of the Fermi Fock space representation of the Clifford algebra on H ⊕ H∗ with quadratic form Q(X + α) := 2α(X) for X ∈ H, α ∈ H∗ .
(3.5)
In the standard irreducible infinite Clifford module F, these are given by the linear action of exterior and interior multiplication by the basis elements {ei }i∈Z for H and their duals {˜ ei }i∈Z for H∗ (3.6) ψi := ei ∧, ψi∗ := ie˜i ∈ End(F). This implies the generating relations of the Clifford algebra [ψi , ψj∗ ]+ := ψi ψj∗ + ψj∗ ψi = δij ,
[ψi , ψj ]+ = [ψi∗ , ψj∗ ]+ = 0.
(3.7)
Normal ordering :O: of any finite product of such creation and annihilation operators is defined by placing all the annihilation factors ψi∗ to the right for i ≥ 0 and all the creation factors ψi to the left for i < 0 (taking the signs following from anticommutativity into account). This implies the vanishing of vacuum expectation values (VEV’s) of all normally ordered operators (other than scalars) h0|:O:|0i = 0.
(3.8)
If the partition λ is expressed in Frobenius notation as (a1 , . . . ar |b1 , . . . , br ), where the ai ’s and bi ’s are strictly decreasing sequences of non-negative integers, denoting the number of boxes to the right of and below the ith diagonal box of the corresponding Young diagram (the “arm” and “leg” lengths), with the Frobenius index r equal to the number of terms on the principal diagonal, we may express the basis element |λ; N i as Pr
|λ; N i = (−1)
i=1 bi
r Y
∗ ψai +N ψ−b |N i. −1+N i
(3.9)
i=1
The Fermi fields ψ(z) and ψ ∗ (z) are defined as generating series for these creation and annihilation operators X X ψ(z) := ψi z i , ψ ∗ (z) := ψi∗ z −i−1 . (3.10) i∈Z
i∈Z
Elements g ∈ GL(H) of the connected group of invertible endomorphisms of H may be viewed as elements of the larger group Spin(H + H∗ , Q) of endomorphisms of H + H∗ preserving the quadratic form Q. If they are represented within the basis {ei }i∈Z by the doubly infinite matrix exponential g ∼ eA (3.11) 14
with elements {Aij }i,j∈Z , they have the Clifford representation X gˆ := exp Aij :ψi ψj∗ :.
(3.12)
i,j∈Z
The Pl¨ ucker map into the projectivization PF of the Fock space Pl : GrH+ (H) → PF Pl : W := span{w1 , w2 , . . . } 7→ [w1 ∧ w2 ∧ · · · ]
(3.13)
(where [. . . ] denotes the projective class) embeds the infinite Grassmannian GrH+ (H) into N PF as the union of the orbits of the subspaces H+ under the Clifford action of GL(H) if W has virtual dimension N . Thus the Pl¨ ucker map intertwines the action of GL(H) on GrH+ (H) and the (projectivized) Clifford representation on F ∀ W ∈ GrH+ (H).
Pl(g(W )) = gˆ(Pl(W )),
(3.14)
In particular, the Clifford representation of the KP and 2D Toda flow groups Γ+ and Γ− is given by P∞ P∞ γˆ+ (t) = e i=1 ti Ji , γˆ− (s) = e i=1 si J−i , (3.15) where the operators Ji are bilinears in the creation and annihilation operators X ∗ Ji := :ψj ψj+i :, i ∈ Z
(3.16)
j∈Z
that satisfy the commutation relations of the infinite Heisenberg algebra [Ji , J−j ] = iδij .
(3.17)
The projective representation may be viewed as a representation of this central extension of the abelian group generated by the elements γ+ (t), γ− (s), with P∞
γˆ+ (t)ˆ γ− (s) = e
i=1
iti si
γˆ− (s)ˆ γ+ (t).
The Ji ’s may be viewed as Fourier components of the current operator X J(z) := Ji z i .
(3.18)
(3.19)
i∈Z
The effect of conjugation of the creation and annihilation operators by a group element is gˆψi gˆ−1 =
X
gˆ−1 ψi∗ gˆ =
ψj gji ,
j∈Z
X j∈Z
15
gij ψj∗
(3.20)
if g(ei ) =
X
ej gji .
(3.21)
j∈Z
In particular, γˆ+ (t)ψ(z)ˆ γ+−1 (t) = eξ(t,z) ψ(z), γˆ− (s)ψ(z)ˆ γ−−1 (s) = eξ(s,z
3.2
−1 )
γˆ+ (t)ψ ∗ (z)ˆ γ+−1 (t) = e−ξ(t,z) ψ ∗ (z),
ψ(z),
γˆ− (s)ψ ∗ (z)ˆ γ−−1 (s) = e−ξ(s,z
−1 )
ψ ∗ (z).
(3.22) (3.23)
Fermionic construction of τ -functions, Baker function and adapted bases
In Sato’s approach to KP τ -functions [40, 41], given an element gˆ of the Fermionic representation of the Clifford algebra that satisfies the bilinear relation. [I, gˆ ⊗ gˆ] = 0,
(3.24)
ψi ⊗ ψi∗ ∈ End(F ⊗ F),
(3.25)
where I :=
X i∈Z
the following vacuum expectation value τg (t) = h0|ˆ γ+ (t)ˆ g |0i
(3.26)
satisfies the Hirota bilinear relation (2.9) and hence is a KP τ -function. In the following, we call such elements group-like elements [8]. In particular, if P
gˆ = e
i,j∈Z
Aij :ψi ψj∗ :
(3.27)
is the fermionic representation of the element g ∈ GL(H) with matrix representation eA in the standard {ei }i∈Z basis, gˆ satisfies (3.24) and τg (t) is the same as the τ -function τW (t) defined in (2.8) with W = gH+ . The Baker function defined in (2.12) can then be expressed as the following VEV [40, 41] h0|ψ0∗ γˆ+ (t)ψ(z)ˆ g |0i , τg (t)
(3.28)
h0|ψ−1 γˆ+ (t)ψ ∗ (z)ˆ g |0i . τg (t)
(3.29)
− Ψ− g (z, t) := ΨW (z, t) =
and the dual Baker function (2.13) is + Ψ+ g (z, t) := ΨW (z, t) =
Since h0|ˆ γ+ (t)|λi = sλ (t) = hλ|ˆ γ− (t)|0i, 16
(3.30)
it follows that the τ -function may, at least formally, be expressed as a sum over the Schur function basis X τg (t) = πλ (g)sλ (t) (3.31) λ
where πλ (g) := hλ|ˆ g |0i
(3.32)
is the λ Pl¨ ucker coordinate of the Grassmannian element W and the Schur functions sλ are viewed as functions of the KP flow variables t = (t1 , t2 , . . . ). More generally, for any such gˆ, we may define a lattice of KP τ -functions generating solutions to the modified KP (mKP) hierarchy, τgmKP (N, t) := hN |ˆ γ+ (t)ˆ g |N i
(3.33)
which satisfy the extended system of Hirota equations (2.19) of the mKP hierarchy. Introducing a second set of flow variables s = (s1 , s2 , . . . ) we may define the 2D Toda lattice of τ -functions τg (N, t, s) := hN |ˆ γ+ (t)ˆ g γˆ− (s)|N i, (3.34) which satisfy I
0
z N −N e−ξ(δt,z) τg (N, t + [z −1 ], s)τg (N 0 , t + δt − [z −1 ], s + δs)dz =
Iz=∞ 0 −1 z N −N e−ξ(δs,z ) τg (N − 1, t, s + [z])τg (N 0 + 1, t + δt, s + δs − [z])dz,
(3.35)
z=0
understood to hold identically in δt = (δt1 , δt2 , . . . ), δs := (δs1 , δs2 , . . . ) and in N , N 0 . In the following, we make the further assumption that the group element g is represented by lower triangular matrices gij = 0 if j > i. (3.36) This means that h0|ˆ g = h0|ˆ g −1 = h0|.
(3.37)
Applying the group element gˆ, we obtain the fermionic VEV respresentation of the bases g∗ , {wk∈Z } defined in (2.26), (2.27).
g {wk∈Z }
Proposition 3.1. The bases defined in eqs. (2.26) , (2.27) have the following representation as fermionic VEV’s ( h0|ψ−k gˆ−1 ψ ∗ (z)ˆ g |0i if k ≥ 1, g wk (z) = (3.38) −1 ∗ h0|ˆ g ψ (z)ˆ g ψ−k |0i if k ≤ 0, ( ∗ h0|ψk−1 gˆ−1 ψ(z)ˆ g |0i if k ≥ 1, g∗ wk (z) = (3.39) −1 ∗ h0|ˆ g ψ(z)ˆ g ψk−1 |0i if k ≤ 0. 17
Proof. Substitute the fourier series representations (3.10) of the fermionic field operators ψ(z), ψ ∗ (z) in the RHS of (3.38) and (3.39) and apply the conjugation relations (3.20) term g g∗ by term. This results in the series representations (2.26), (2.27) for {wk∈Z } , {wk∈Z }, both for k ≥ 1 and k ≤ 0. Now define the sequence of subspaces WNg , WNg∗ , N ∈ Z g N (H) WNg = span{w−N +k }k∈N+ ∈ GrH+
(3.40)
g∗ WNg⊥ = span{wN +k }k∈N+ ∈ GrH−N (H),
(3.41)
+
which coincide with (2.20), (2.22). The charge N sector Baker function and its dual ∗ hN |ψN γˆ+ (t)ψ(z)ˆ g |N i ∈ WNg⊥ τg (t) hN |ψN −1 γˆ+ (t)ψ ∗ (z)ˆ g |N i Ψ+ ∈ WNg g (z, N, t) := τg (t)
Ψ− g (z, N, t) :=
(3.42) (3.43)
can be expanded in these bases. For the case N = 0, we have Proposition 3.2. Ψ− g (z, t)
=
Ψ+ g (z, t) =
∞ X j=1 ∞ X
αjg (t)wjg∗ (z)
(3.44)
βjg (t)wjg (z),
(3.45)
j=1
where αjg (t) :=
g ψj−1 |0i h0|ψ0∗ γˆ+ (t)ˆ , τg (N, t)
βjg (t) :=
∗ |0i h0|ψ−1 γˆ+ (t)ˆ g ψ−j . τg (N, t)
(3.46)
Proof. In eq. (3.28), insert I = gˆgˆ−1 and the projection operator to the N = 1 sector F1 ⊂ F of the fermionic Fock space X Π1 := |λ; 1ihλ; 1|, (3.47) λ
(where the sum is over all partitions λ). This gives 1 X Ψ− (z, t) = h0|ψ0∗ γˆ+ (t)ˆ g |λ; 1ihλ; 1|ˆ g −1 ψ(z)ˆ g |0i. g τg (t) λ
(3.48)
In this sum, the only nonvanishing terms are the trivial partition λ = ∅ and the horizontal partitions λ = (j), j = 1, 2, . . . . To see this, use (3.20) to write the second factor in the sum (3.48) as ∞ X X hλ; 1|ˆ g −1 ψ(z)ˆ g |0i = zj hλ; 1|ψi |0igij−1 . (3.49) j∈Z
18
i=j
Expressing the partition λ in Frobenius notation as (a1 , . . . , ar |b1 , . . . , br ), the state |λ; 1i is, by (3.9) r Y Pr ∗ |λ; 1i = (−1) i=1 bi ψai +1 ψ−b ψ0 |0i. (3.50) i i=1
It follows from the fermionic Wick theorem that hλ; 1|ψi |0i =
h0|ψ0∗
r Y
ψ−bi ψa∗i +1 ψi |0i
(3.51)
i=1
vanishes unless either λ = ∅ or r = 1 and b1 = 0. For the first case, we have h∅; 1|ˆ g −1 ψ(z)ˆ g |0i = w1g∗ (z),
(3.52)
for the other case λ = (j), we have g∗ (z), h(j); 1|ˆ g −1 ψ(z)ˆ g |0i = wj+1
j = 1, 2, . . .
(3.53)
by eq. (2.27), and h0|ψ0∗ γˆ+ (t)ˆ g |(j − 1); 1i = τg (t)αjg (t),
(3.54)
proving eq.(3.44). Eq. (3.45) is proved similarly.
3.3
Multiplicative recursion relations for general gˆ
Multiplying the adapted basis elements by z, it follows that they satisfy the recursion relations Proposition 3.3. zwkg (z)
=
k+1 X
˜ + wg (z) Q kj j
(3.55)
˜ − wg (z), Q kj j
(3.56)
j=−∞
zwkg∗ (z)
=
k+1 X j=−∞
where ˜+ = Q kj
−j X
−1 ˜− g−j,i gi+1,−k = Q 1−j,1−k ,
j ≤k+1
(3.57)
j ≤ k + 1.
(3.58)
i=−k−1
˜− = Q kj
k−1 X
−1 ˜+ gk−1,i gi+1,j−1 = Q 1−j,1−k ,
i=j−2
19
Proof. This can either be proved directly, by inverting the Fourier series expansions for wkg (z) and wkg∗ (z), or by using the fermionic identities zψ(z) = [J1 , ψ(z)],
zψ ∗ (z) = −[J1 , ψ ∗ (z)]
(3.59)
and evaluating the RHS of the fermionic representations (3.38) and (3.39) for both wkg (z) and wkg∗ (z) by introducing a sum over a complete set of intermediate states in the N = 0 or N = −1 fermionic charge sectors; i.e., by inserting the projection operator X ΠN := |λ; N ihλ; N | (3.60) λ
onto these sectors. ˜ + may be split into The recursion matrix Q ˜ −− Q ˜+ = · · · Q ˜ +− Q
block form .. ˜ −+ . Q ··· , .. ˜ ++ . Q
(3.61)
where the range of indices are: ˜ −− : k ≤ 0, l ≤ 0; Q kl ˜ +− : k ≥ 1, l ≤ 0; Q kl
˜ −+ : k ≤ 0, l ≥ 1; Q kl ˜ ++ : k ≥ 1, l ≥ 1. Q kl
(3.62)
It follows from the above computation that the fermionic representation of the recursion matrix elements is Corollary 3.4. ˜ −− Q kl ˜ Q−+ kl +− ˜ Qkl ˜ ++ Q kl where
∗ −1 = h0|ψ−l gˆ J1 gˆψ−k |0i − δkl κ+ , k ≤ 0, l ≤ 0 −1 = δk0 δl1 g00 g−1,−1 , k ≤ 0, l ≥ 1 ∗ −1 = h0|ψ−k ψ−l gˆ J1 gˆ|0i, k ≥ 1, l ≤ 0 ∗ |0i + δkl κ− , k ≥ 1, l ≥ 1, = −h0|ψ−k gˆ−1 J1 gˆψ−l
∞ X −1 −1 κ+ := (gj+1,j gjj + gjj gj,j−1 ),
κ− :=
j=0
−1 X
−1 −1 (gj+1,j gjj + gjj gj,j−1 ).
(3.63)
(3.64)
j=−∞
˜ −+ has just one nonzero entry, in the lower triangular corner (01), and both Note that Q kl ˜ −− and Q ˜ ++ are nearly lower triangular, with only a single nonzero diagonal immediately Q kl kl above the principal one.
20
Remark 3.1. Proposition 3.3 provides a system of recursion relations satisfied by the adapted ˜ ± are almost basis elements {wkg (z)}, {wkg∗ (z)} in the sense that the recursion matrices Q lower triangular, with a single nonvanishing diagonal above the principle one. This means g that, given all the preceding {wjg (z)}’s, for j ≤ k, we can uniquely determine wk+1 (z) (and g∗ similarly for wk+1 (z)). This is only a finite recursion system in the case when the matrices ˜ ± are of finite band width below the principle diagonal. In Section 4 it will be shown that Q this is indeed the case when we specialize to group elements of the convolution type, with a generating function G(z) that is polynomial, and the number of nonvanishing parameters si in the abelian group element γˆ− (s) finite.
3.4
Euler derivative relations for general gˆ
Applying the Euler operator d dz g g∗ to the adapted bases {wk (z)}, {wk (z)} gives the following linear relations D := z
(3.65)
Proposition 3.5. Dwkg (z)
=
k X
+ g P˜kj wj (z)
(3.66)
− g∗ P˜kj wj (z)
(3.67)
j=−∞
Dwkg∗ (z)
=
k X j=−∞
where the lower triangular matrices P˜ ± have as entries + P˜kj =
k X
−1 ig−j,−i g−i,−k − δjk ,
j ≤ k,
(3.68)
i=j − P˜kj =
k X
−1 igk−1,i−1 gi−1,j−1 − δjk
j ≤ k.
(3.69)
i=j
Proof. This can again be proved either directly, by inverting the Fourier series expansions for wkg (z) and wkg∗ (z), or by using the fermionic identities ˆ ψ(z)], Dψ(z) = [H,
ˆ ψ ∗ (z)] − ψ ∗ (z), Dψ ∗ (z) = −[H,
(3.70)
ˆ is the fermionic energy operator where H ˆ := H
X
j :ψj ψj∗ :,
(3.71)
j∈Z
and again evaluating the RHS for the fermionic expressions by introducing the projector ΠN to the N = 0 or N = −1 fermionic charge sectors. 21
The differential recursion matrix P˜ + may again be split into block form . P˜ −− .. P˜ −+ = 0 ··· P˜ + = · · · . P˜ +− .. P˜ ++
(3.72)
where the range of indices are again: P˜kl−− : k ≤ 0, l ≤ 0; P˜kl+− : k ≥ 1, l ≤ 0;
P˜kl−+ : k ≤ 0, l ≥ 1; P˜kl++ : k ≥ 1, l ≥ 1.
(3.73)
Proposition 3.6. The fermionic representation of the Euler differential matrix elements is P˜kl−− P˜kl−+ P˜kl+− P˜ ++ kl
∗ −1 ˆ = −h0|ψ−l gˆ H gˆψ−k |0i, k ≤ 0, l ≤ 0 = 0, k ≤ 0, l ≥ 1 ˆ gˆ|0i − δk,−l , k ≥ 1, l ≤ 0 = h0|ψ−l ψk∗ gˆ−1 H −1 ˆ ∗ = −h0|ψ−k gˆ H gˆψ−l |0i, k ≥ 1, l ≥ 1.
(3.74)
Note that both P˜kl−− and P˜kl++ are lower triangular. Proof. A direct evaluation of the matrix elements. Remark 3.2. Proposition 3.6 provides a recursive system of linear differential relations satisfied by the adapted basis elements {wkg (z)}, {wkg∗ (z)} in the sense that the matrices P˜ ± are lower triangular. This means that, given all the {wjg (z)}’s, for j ≤ k, we can uniquely determine Dwkg (z) (and similarly for Dwkg∗ (z)). This is again only a finite system in the case when the matrices P˜ ± are of finite band width below the principal diagonal. In Section 4 it will be shown that this is indeed the case when we require that the number of nonvanishing parameters si in the abelian group element γˆ− (s) be finite.
3.5
The n-pair correlation function and Christoffel-Darboux type kernel
The n-pair correlation function is defined by the following VEV g ˜ 2n K (z1 , . . . , zn ; w1 , . . . , wn ) := h0|
n Y
(ψ(zi )ψ ∗ (wi )) gˆ|0i,
(3.75)
i=1
provided |zi | > |wi | ∀i. Adding the KP multitime flow variables gives the time dependent correlation function g ˜ 2n K (z1 , . . . , zn ; w1 , . . . , wn , t) := h0|ˆ γ+ (t)
n Y i=1
22
(ψ(zi )ψ ∗ (wi )) gˆ|0i.
(3.76)
In particular, for n = 1, t = 0, ( h0|ψ(z)ψ ∗ (w)ˆ g |0i if |z| > |w|. g ˜ K2 (z; w) := −h0|ψ ∗ (w)ψ(z)ˆ g |0i if |z| < |w|.
(3.77)
This is equivalent to the following expression in terms of the τ -function Proposition 3.7. −1 −1 ˜ 2g (z, w) = τg ([w ] − [z ]) . K z−w
(3.78)
Proof. See Appendix A.1 g ˜ 2n By Wick’s theorem, K may be expressed as a determinant in terms of the 2-point function
Lemma 3.8. g ˜ 2n ˜ 2g (zi , wj , t)|1≤i,j≤n K (z1 , . . . , zn ; w1 , . . . , wn , t) = det(K
(3.79)
g ˜ 2n To express the multipair correlator K in terms of the τ -function it is helpful to introduce the following notation.
Definition 3.1. For a set of n nonzero complex numbers z = (z1 , z2 , . . . zn ), we use the notation [z−1 ] to denote the infinite sequence {[z]i }i=1,2,··· , of normalized power sums n
−1
−1
−1
[z ] := ([z ]1 , [z ]2 , . . . ),
1X 1 [z ]i := . i j=1 (zj )i −1
(3.80)
Lemma 3.9. For 2n distinct nonzero complex parameters {zi , wi }i=1,...n provided |zi | > |wi | ∀i, we have h0|
n Y
∗
(ψ(zi )ψ (wi )) = det
i=1
1 zi − wj
h0|ˆ γ+ ([w−1 ] − [z−1 ]).
(3.81)
1≤i,j≤n
Proof. See Appendix A.2. It follows that, for an arbitrary group element g ∈ GL(H), we can express the 2n-point time dependent pair correlation function (3.77) in terms of the τ -function. Corollary 3.10. g ˜ 2n K (z, w, t) =
n Y n ξ(t,zj )−ξ(t,wj ) −1 −1 (e )τg (t − [z ] + [w ])deti,j=1 j=1
23
1 zi − wj
(3.82)
Proof. (See Appendix A.2.) It is first proved for n = 1, by applying Lemma 3.9 for n = 1 to deduce h0|ˆ γ+ (t)ψ(z)ψ ∗ (w)ˆ g |0i = eξ(t,z)−ξ(t,w) h0|ψ(z)ψ ∗ (w)ˆ γ+ (t)ˆ g |0i eξ(t,z)−ξ(t,w) h0|ˆ γ+ (t − [z −1 ] + [w−1 ])ˆ g |0i , (3.83) = z−w where we have used eq. (3.22) and then extended to arbitrary n, either by induction or, equivalently, by the fermionic Wick theorem. Using Wick’s theorem, we have the following determinantal expression in terms of single pair correlators and in terms of the τ -function. Corollary 3.11. g ˜ 2n K (z, w, t)
=
detni,j=1
eξ(t,zi )−ξ(t,wj ) τg (t − [zi −1 ] + [wj −1 ]) zi − wj
.
(3.84)
From the fermionic representation, it is also easy to see that the 2-point function at t = 0 may be expressed as a series Proposition 3.12. ˜ 2g (z, w) K
(P ∞ g g∗ j=1 wj (w)w−j+1 (z) if |z| > |w|, = P g g∗ − ∞ j=1 w−j+1 (w)wj (z) if |z| < |w|.
(3.85)
Proof. A fermionic proof is given in Appendix A.3. Alternatively, eq. (3.85) may be proved using the n = 1 case of (3.11) and the Schur function expansion (3.31) of the τ -function. We also have the following Christoffel-Darboux type formula. Proposition 3.13. P∞ P∞ ˜ 2g (z, w) = K
i=0
g∗ ˜ g j=0 (Aij w−j (z)w−i (w))
g g∗ ˜+ −Q 01 w1 (z)w1 (w)
z−w
,
(3.86)
where ˜+ A˜ij = Q j+1,−i ,
(3.87)
˜ + is the recursion matrix appearing in eq. (3.57). and Q ij Proof. Multiply the RHS of eq. (3.85) by z − w. Then the result follows from using the recursion relations (3.55) directly, together with the telescopic cancellation between the two types of terms coming from the z and w recursion relations. Alternatively, it may be derived directly from the VEV representation eq. (3.76) by using the fermionic identity (z − w)ψ(w)ψ ∗ (z) = [J1 , ψ(w)ψ ∗ (z)]
(3.88)
and inserting sums over complete sets of intermediate states in both terms arising from the commutator. 24
3.6
Multicurrent correlator for general gˆ
In the following, it will be convenient to use the inverse variables xi := 1/zi in expressing the correlators. Definition 3.2. We denote the positive frequency part of the current operator as ∞ X
J+ (x) :=
xi J i ,
(3.89)
i=1
and define the multicurrent correlator Jn (x1 , . . . , xn ) as n Y
Jn (x1 , . . . , xn ) := h0|
! J+ (xi ) gˆ|0i.
(3.90)
i=1
We also introduce the correlators Wn (x1 , . . . , xn ), defined as multiple derivatives of the τ -function ! n Y Wn (x1 , . . . , xn ) := ∇(xi ) τg (t) , (3.91) i=1
t=0
where the commuting parametric family of first order gradient operators ∇(xi ) in the tvariables is defined by ∞ X ∂ ∇(x) := xi−1 . (3.92) ∂ti i=1 For later purposes, we also introduce a “connected” version of these quantities, defined by replacing τg by ln(τg ) ! n Y ˜ n (x1 , . . . , xn ) := W ∇(xi ) (ln(τg (t)) . (3.93) i=1
t=0
Assuming τg (t) to be normalized such the τg (0) = 1, we have, in particular, ˜ 1 (x1 ) = W1 (x1 ) W ˜ 2 (x1 , x2 ) = W2 (x1 , x2 ) − W1 (x1 )W2 (x2 ) W ˜ 3 (x1 , x2 , x3 ) = W3 (x1 , x2 , x3 ) − W1 (x1 )W2 (x2 , x3 ) − W1 (x2 )W2 (x1 , x3 ) − W1 (x3 )W2 (x1 x2 ) W + 2W1 (x1 )W1 (x2 )W1 (x3 ) .. .. . = . (3.94) From the fermionic formula (3.26) for τg (t) it follows that Wn and the multicurrent correlator Jn are related by Lemma 3.14. Jn (x1 , . . . , xn ) =
n Y
! xi
Wn (x1 , . . . , xn ).
(3.95)
i=1
Proof. This is immediate from applying (3.91) to the fermionic formula (3.26) for τg (t). 25
4 4.1
Specialization to the case of hypergeometric τ -functions Convolution products
An abelian group that is central to hypergeometric τ -functions is obtained by extending the semigroup C := {Cρ } consisting of formal convolution products with functions or distributions on the circle S 1 admitting a Fourier series representation ∞ X
ρ(z) =
ρi z −i−1
(4.1)
i=−∞
i.e. the diagonal action on an element w ∈ H with Fourier representation ∞ X
w(z) :=
wi z −i−1 ,
(4.2)
i=−∞
consisting of multiplication of their Fourier coefficients ∞ X
Cρ (w)(z) :=
ρi wi z −i−1 ,
(4.3)
i=−∞
to include the diagonal action of commuting differential operators of the form ! X exp αi D i ,
(4.4)
i
where D is the Euler operator (3.65) (or simply the vector field of infinitesimal rotations on the circle). (These may also formally be represented as a convolution with the series ρ(z) in P j which the coefficients in (4.1) are ρi = exp j αj (−i − 1) .) In the L2 (S 1 ) sense, the convolution product may be viewed as the integral I I z dζ 1 z dζ 1 ρ(ζ)w = ρ (4.5) w(ζ) . Cρ (w)(z) := 2πi S 1 ζ ζ 2πi S 1 ζ ζ H 1 Alternatively, in the formal Laurent series sense, the integral 2πi f (ζ) dζζ may be reinterS1 preted as a formal residue at z = 0. These are essential to the definition of KP or 2D-Toda τ -functions of hypergeometric type [37, 38], since they correspond to Grassmannian elements of the form W (ρ,s) := Cρ γ− (s)(H+ ) ∈ GrH+ (H)
(4.6)
for the zero virtual dimension component or, more generally (ρ,s)
WN
= Cρ γ− (s)(z −N H+ ), 26
N ∈Z
(4.7)
in the virtual dimension N sector. In the following, the coefficients ρi =: eTi
(4.8)
in the distributional (or formal) series ρ(z) will always be understood as determined by the weight generating function G and the constants (β, γ), as in [22], such that (G,β)
ri
= G(iβ) =
ρi . γρi−1
(4.9)
Since the τ -function is only projectively defined, assuming ρ0 6= 0, we may choose it as ρ0 = 1 and express the coefficients ρj as finite products of the γG(iβ)’s and their inverses [22] ρj = γ
j
j Y
G(iβ),
ρ−j = γ
i=1
−j
j−1 Y (G(−iβ))−1 ,
j = 1, 2, . . . .
(4.10)
i=0
Thus Tj (β, γ) := jln(γ) +
j X
ln(G(iβ)),
j = 1, 2, . . . ,
T0 := 0
i=1 j−1
T−j (β, γ) := −jln(γ) −
X
ln(G(−iβ)),
j = 1, 2, . . .
(4.11)
i=0
Although vanishing values ρj = 0 for some negative coefficient j = −n − 1 are allowed, the relations (4.10) then imply that ρj = 0 ∀ i ≤ −n − 1 (G,β)
and hence all the coefficients γ |λ| rλ partitions of length `(λ) > n.
4.2
(4.12)
appearing in the τ -functions (1.26), (1.27) vanish for
Adapted basis, τ -function and Baker function for the hypergeometric case
Since γ− (s) is the generating function for the complete symmetric functions, γ− (s) =
∞ X
hi (s)z −i ,
(4.13)
i=0
specializing to the case g := Cρ γ− (β −1 s),
(4.14)
we have gij = ρi hi−j (β −1 s),
−1 gij−1 = ρ−1 s). j hi−j (−β
27
(4.15)
(G,β,γ,s)
The corresponding adapted bases {wk (z)}k∈Z defined by applying the element −1 Cρ γ− (β s) and its dual to the basis elements {em := z −m−1 }m∈Z are (G,β,γ,s) wk (z)
:= Cρ (γ− (β
−1
s)z
k−1
k−1 X
)=
hk−j−1 (β −1 s)ρ−j−1 z j ,
(4.16)
j=−∞
where the coefficients ρj are defined in eq. (4.10). The dual basis, under the Hirota bilinear pairing (2.11), is defined by (G,β,γ,s)∗ wk (z)
:=
ιCρ−1 (γ+ (−β −1 s)ι(z k−1 ))
=
k−1 X
j hk−j−1 (−β −1 s)ρ−1 j z ,
(4.17)
j=−∞
where ι is the involution (2.23). Simplifying the notation for the adapted bases in this case to (G,β,γ,s) (G,β,γ,s)∗ wk (z) =: wk (z), wk (z) =: wk∗ (z) (4.18) these have the fermionic representation (cf. (3.38)) ( h0|ψ−k γˆ−−1 (s) Cˆρ−1 ψ ∗ (z)Cˆρ γˆ− (β −1 s)|0i if k ≥ 1, wk (z) := h0|ψ ∗ (z)Cˆρ γˆ− (β −1 s)ψ−k |0i if k ≤ 0,
(4.19)
which coincides with the basis defined in (4.16), and the dual basis, defined by (3.39), becomes ( ∗ h0|ψk−1 γˆ− (−β −1 s)Cˆρ−1 ψ(z)Cˆρ γˆ− (β −1 s)|0i if k ≥ 1 (4.20) wk∗ (z) := ∗ h0|ψ(z)Cˆρ γˆ− (β −1 s)ψk−1 |0i if k ≤ 0, which coincides with (4.17). For k ∈ N+ they define corresponding elements of the Grassmanian: W (G,β,γ,s) = span{wk (z)}k∈N+
(4.21)
span{wk∗ (z)}k∈N+
(4.22)
W
(G,β,γ,s)⊥
=
The fermionic representation of the group element associated to the weight generating function γG defined in (4.10) is of the form gˆ := Cˆρ γˆ− (β −1 s)
(4.23)
ˆ Cˆρ = eA
(4.24)
where with the algebra element Aˆ is Aˆ :=
X
Ti :ψi ψi∗ :
i∈Z
28
(4.25)
τ
We therefore have the following fermionic representation of the 2D Toda τ -function (t, s) at N = 0 defined by (1.26)
(G,β,γ)
τ (G,β,γ) (t, s) = h0|ˆ γ+ (t)Cˆρ γˆ− (s)|0i.
(4.26)
. Remark 4.1. Such 2D Toda τ -functions, referred to in [37, 38] as τ -functions of hypergeometric type, were considered, for other purposes, in [30], but without identification of these as Hurwitz generating functions. Fermionic representations of particular families of the Hurwitz numbers were considered in [5, 7, 20–22, 24, 33–38, 46] Viewed as a parametric family of KP τ -functions, after the substitution s → β −1 s, the corresponding Baker function and its dual have the fermionic representation −1 Ψ− s) = (G,β,γ) (z, t, β −1 Ψ+ s) = (G,β,γ) (z, t, β
h0|ψ0∗ γˆ+ (t)ψ(z)Cˆρ γ− (β −1 s)|0i , τ (G,β,γ) (t, β −1 s) h0|ψ−1 γˆ+ (t)ψ ∗ (z)Cˆρ γ− (β −1 s)|0i τ (G,β,γ) (t, β −1 s)
(4.27) .
(4.28)
As particular cases of the expansions (3.44), (3.45), the Baker function and its dual for this class of hypergeometric τ -functions can be expressed relative to the dual bases {wk∗ (z)}, {wk (z)} as −1 Ψ− s) (G,β,γ) (z, t, β
=
−1 Ψ+ s) = (G,β,γ) (z, t, β
∞ X k=1 ∞ X
(G,β,γ)
(t, s),
(4.29)
(G,β,γ)
(t, s),
(4.30)
wk∗ (z)αk wk (z)βk
k=1
where h0|ψ0∗ γˆ+ (t)Cˆρ γˆ− (β −1 s)ψk−1 |0i := , τg (t, β −1 s) ∗ h0|ψ−1 γˆ+ (t)Cˆρ γˆ− (β −1 s)ψ−k |0i (G,β,γ) βk (t, s) := . τg (t, β −1 s)
(G,β,γ) αk (t, s)
4.3
(4.31) (4.32)
An alternative realization of convolution actions
It is easy to see that for (4.10) the convolution operator can formally be equivalently realized by the exponential of a differential operator, namely Cρ = eT (−D−1) 29
(4.33)
where the T (x) is viewed as a formal Taylor series satisfying eT (x)−T (x−1) = γG(βx),
(4.34)
and T (0) = 0 with T (i) = Ti
for i ∈ Z.
(4.35)
This series, though nonunique, may be given an explicit expression as T (x) =
∞ X
(−1)k+1 β k Ak pk (x),
(4.36)
k=0
where A0 = − log γ, ∞ X 1 ck , k > 0, Ak = k j=1 j
(4.37) (4.38)
the polynomials {pk (x) ∈ x C[x]} are determined by pk (x) − pk (x − 1) = xk
(4.39)
and are closely related to the Bernoulli polynomials. In particular, p0 (x) = x, 1 x (x + 1) , p1 (x) = 2 1 p2 (x) = x (x + 1) (2 x + 1) , 6 1 2 x (x + 1)2 , p3 (x) = 4 1 p4 (x) = x (x + 1) (2 x + 1) 3 x2 + 3 x − 1 , 30 .. .
(4.40) (4.41) (4.42) (4.43) (4.44) (4.45)
Their generating function is ∞
X pk (x)ak 1 ax (e − 1). = . 1 − e−a k! k=0
(4.46)
Remark 4.2. There are, of course, an infinity of other series T (x) satisfying eqs. (4.34) and (4.35), obtained, e.g. by adding any periodic function P (x) of unit period satisfying P (0) = 0. 30
The fermionic operator Aˆ appearing in eqs. (4.25), may equivalently be expressed as Aˆ = resz=0 (:ψ(z)T (−D − 1) ψ ∗ (z):) ,
(4.47)
which satisfies the commutation relations h i ˆ ψ(z) = T (D) ψ(z), A, h i ∗ ˆ A, ψ (z) = −T (−D − 1) ψ ∗ (z).
(4.48) (4.49)
The adapted basis elements can equivalently be expressed as wk (z) = eT (−D−1) γ− (β −1 s)z k−1 ,
(4.50)
wk∗ (z) = e−T (D) γ− (−β −1 s)z k−1 .
(4.51)
The linearly independent elements {wN +k }k∈N+ span an infinite nested sequence of subspaces (G,β,γ,s) WN := span{wN +k }k∈N+ (4.52) of virtual dimension N (G,β,γ,s)
· · · ⊂ WN −1
(G,β,γ,s)
⊂ WN
(G,β,γ,s)
⊂ WN +1
⊂ ··· .
(4.53)
This defines an infinite complete flag which, for fixed parameters s and convolution elements ρ, determines a chain of τ -functions of the mKP hierarchy, defined as determinants of the corresponding orthogonal projection maps τ (G,β,γ) (N, t, β −1 s) := τW (G,β,γ,s) (N, t)
(4.54)
N
=
X
(G,β)
γ |λ| r(λ,N ) sλ (t)sλ (β −1 s),
(4.55)
λ (G,β)
where the content product coefficient rλ (G,β) r(λ,N ) :=
, in eq. (1.26) is replaced by the shifted version
Y
G(β(N + j − i)).
(4.56)
(i,j)∈λ
Alternatively, viewing the parameters s = (s1 , s2 . . . ) as a further infinite set of flow variables, this defines a hypergeometric τ -function of the 2D Toda hierarchy. Viewing all parameters {N, t, s} on the same footing, as discrete or continuous flow parameters, we obtain a lattice of τ -functions of the 2D-Toda hierarchy.
31
4.4
Recursion relations in the hypergeometric case
We also introduce an alternative notation for these bases which is used in refs. [1, 2] Ψ+ k (x) := γw1−k (z = 1/x),
∗ Ψ− k (x) := w1−k (z = 1/x),
(4.57)
where
1 (4.58) x= . z The basis elements {Ψ± k (x)} defined in (4.57) are viewed as elements of the subspaces ± −1 −1 W(G,β,γ,s) := span{Ψ± ]((γ)) −k (x)}k∈N ⊂ K[x, x , s, β, β
(4.59)
in the formal series sense. They may be understood as elements of formal power series analogs of the Sato-Segal-Wilson Grassmannian, consisting of subspaces of the Hilbert space spanned by powers {z k }k∈Z of the spectral parameter z = 1/x, commensurable with the subspace spanned by the monomials {z k }k=0,1,... . −1 −1 The Baker function Ψ− s) and its dual Ψ+ s), are (G,β,γ) (z = 1/x, t, β (G,β,γ) (z = 1/x, t, β given by the Sato formula ± Ψ∓ (G,β,γ) (z, t, s) = e
P∞
i=1 ti z
i
P∞ τ (G,β,γ) (t ∓ [x], s) i = e± i=1 ti z (1 + O(1/z)) , (G,β,γ) τ (t, s)
(4.60)
± These may be viewed as elements of the subspaces W(G,β,γ,s) for all values of the KP flow parameters t. − Specializing eqs. (3.55, 3.56) to this case, the basis elements {Ψ+ i (x), Ψi (x)} satisfy the recursion relations
Proposition 4.1. ∞ X 1 + + Ψ = Q+ ij Ψj , γx i j=i−1
(4.61)
∞ X 1 − − Ψ = Q− ij Ψj , γx i j=i−1
(4.62)
˜∓ where the (nearly) upper triangular matrices Q± are the negatives of the transposes of Q −1 ˜ + Q− Qjk . kj = γ
−1 ˜ − Qjk , Q+ kj = γ
(4.63)
In this case we can express them explicitly as Q+ ij
:=
j X
rkG (β)hk−i+1 (β −1 s)hj−k (−β −1 s),
j ≥i−1
(4.64)
k=i−1
Q− ij :=
j X
G r−k (β)hj−k (β −1 s)hk−i+1 (−β −1 s),
k=i−1
32
j ≥ i − 1.
(4.65)
Applying the Euler operator D to the basis elements {wk , wk∗ }, we obtain the following differential linear relations Proposition 4.2. Dwk (z) =
X
+ P˜kj wj (z)
(4.66)
Dwk∗ (z) =
X
− ∗ P˜kj wj (z),
(4.67)
where the elements of the lower triangular matrices P˜ ± are ( (i − 1)δij ∓ β −1 (i − j)si−j , i ≥ j P˜ij± = 0, i < j.
(4.68)
Proof. To see this, we may either use the general formulae (3.69) of Section 3.4, specialized to the case (4.15) or, more simply, proceed directly. Namely, from (4.51) we have Dwk (z) = eT (−D−1) Dγ− (β −1 s)z k−1
(4.69)
= eT (−D−1) (k − 1 − β −1 S(z −1 ))γ− (β −1 s)z k−1 ∞ X −1 = (k − 1)wk (z) − β jsj wk−j (z),
(4.70)
j=1
where S(z) :=
∞ X
ksk z k
(4.71)
k=1
and similarly, Dwk∗ (z)
= (k −
1)wk∗ (z)
+β
−1
∞ X
∗ jsj wk−j (z).
(4.72)
j=1
This linear action is represented by the matrices P˜ ± defined in (4.68). − In terms of the notation {Ψ+ k (x)}, {Ψk (x)}, the differential linear relations are
βDΨ+ k (x)
=
k+1 X
+ + Pkj Ψj (x)
(4.73)
− − Pkj Ψj (x).
(4.74)
j=−∞
βDΨ− k (x)
=
k+1 X j=−∞
where D is the Euler operator in the x-variable D := x
d = −D. dx 33
(4.75)
and the upper triangular matrices P ± are the negatives of the transposes of P˜ ∓ Pij± = −β P˜ji± , ( β(1 − i)δij ∓ (i − j)sj−i , = 0, i > j.
4.5
i≤j
(4.76)
The spectral curve and Kac-Schwarz operators
4.5.1
The quantum and classical spectral curve
Now define the differential operator R acting either on H, or formally, on C[[z, z −1 ]], as: γ (4.77) R := G(−βD), z which acts on the monomial z i : R(z i ) =
ρ−i i−1 γ z . G(−iβ)z i = z ρ−i−1
(4.78)
Applying R to the series for ρ(z), we see this is an eigenvector with eigenvalue 1 R ρ(z) = ρ(z).
(4.79)
R wk = wk−1
(4.80)
(R)j wk = wk−j .
(4.81)
γ G(βD), z
(4.82)
Applying it to wk gives or iterating, Similarly, defining R∗ := we have ∗ R∗ wk∗ = wk−1
(4.83)
∗ (R∗ )j wk∗ = wk−j .
(4.84)
or iterating, It is clear that the operators R and R∗ are invertible. Equivalently, in terms of the x = 1/z variables, we have the operators R+ := γxG(βD) = R,
R− = γxG(−βD) = R∗
(4.85)
and eqs.(4.81), (4.84) are equivalent to + (R+ )j Ψ+ k = Ψk+j
(4.86)
− (R− )j Ψ− k = Ψk+j
(4.87)
Using (4.81) and (4.84), we may re-express eqs. (4.70), (4.72) more compactly as 34
Proposition 4.3. (βD + S(R)) wk = (k − 1)βwk ,
(4.88)
(βD − S(R∗ )) wk∗ = (k − 1)βwk∗ .
(4.89)
or, equivalently, + [βD − S(R+ )] Ψ+ k (x) = kΨk (x),
(4.90)
− [βD + S(R− )] Ψ− k (x) = kΨk (x).
(4.91)
+ ∗ Remark 4.3 (Quantum and classical spectral curves). Since Ψ− 0 (x) = w1 (z), Ψ0 (x) = γw1 (z) are the t = 0 evaluation of the Baker function and its dual, respectively, the k = 0 case of eqs. (4.90) and (4.91)
[βD − S(γxG(+βD))] Ψ+ 0 (x) = 0, [βD +
S(γxG(−βD))] Ψ− 0 (x)
= 0,
(4.92) (4.93)
is satisfied by the Baker function and its dual at t = 0. These are the quantum spectral curve appearing in refs. [1, 2], and its dual. Taking the classical limit amounts to replacing βD by xy, where y is the classical variable canonically conjugate to x. Thus, the classical spectral curve is xy = S(γxG(xy)),
(4.94)
and the dual is the same. 4.5.2
Kac-Schwarz operators
Definition 4.1. If for a given g, an operator a acting on C[[z, z −1 ]] (e.g. a differential N operator in z) stabilizes the element WNg = g(H+ ) of the Sato Grassmannian (resp. the dual g⊥ Grassmannian element WN ) aWNg ⊂ WNg (4.95) or aWNg⊥ ⊂ WNg⊥ ,
(4.96)
we call it a Kac-Schwarz operator (resp. dual Kac-Schwarz operator) for the corresponding KP τ -function [28]. Remark 4.4. Particular examples of Kac-Schwarz operators for weighted Hurwitz numbers were considered in [4, 5].
35
Definition 4.2. Define the following operators: a := eT (−1−D) γ− (β −1 s)zγ− (−β −1 s)e−T (−1−D) = eT (−1−D) ze−T (−1−D) 1 1 = , = z γG(β(−1 − D)) R ∗ −T (D) −1 a := e γ− (−β s)zγ− (β −1 s)eT (D) = e−T (D) zeT (D) 1 1 = z = ∗, γG(β(1 + D)) R
(4.97) (4.98) (4.99) (4.100)
From (4.51) we see that a wk = wk+1
(4.101)
∗
(4.102)
a
wk∗
=
∗ wk+1 ,
(G,β,γ,s)
so these are Kac-Schwarz (resp. dual Kac-Schwarz) operators, since they stabilize WN (G,β,γ,s)⊥ (and WN ) for all N . Similarly, define Definition 4.3.
b := eT (1−D) γ− (β −1 s)Dγ− (−β −1 s)e−T (1−D) = eT (1−D) D − β −1 S(z −1 ) e−T (1−D) (4.103) = D − β −1 S(R)
(4.104)
b∗ := e−T (D) γ− (−β −1 s)Dγ− (β −1 s)eT (D) = e−T (D) D + β −1 S(z −1 ) eT (D)
= D + β −1 S(R∗ ).
(4.105) (4.106)
Then (4.88) and (4.89) are equivalent to b wk = (k − 1)wk ,
(4.107)
∗
(4.108)
b
wk∗
= (k −
1)wk∗ ,
(G,β,γs)
so these too are Kac-Schwarz operators for WN Moreover, for the operators
(G,β,γs)⊥
(and WN
c := a−1 b = RD − β −1 RS(R) c
∗
:= a
∗−1 ∗
∗
b =R D+β
−1
∗
)
(4.109) ∗
R S(R ),
(4.110)
we have c wk = (k − 1)wk−1
(4.111)
∗ c∗ wk∗ = (k − 1)wk−1 .
(4.112)
36
Thus, these also are Kac-Schwarz (and dual Kac-Schwarz) operators for W (G,β,γ,s) and W (G,β,γ,s)⊥ . They satisfy the canonical commutation relations [c, a] = [c∗ , a∗ ] = 1
(4.113)
and completely specify the point of the Grassmanian. More generally, for any N ∈ Z, the operators N , a N := c∗ − ∗ , a
cN := c −
(4.114)
c∗N
(4.115)
give cN wk = (k − 1 − N )wk−1 ,
(4.116)
c∗N
(4.117)
wk∗
= (k − 1 −
∗ N )wk−1 ,
satisfy the canonical commutation relations [cN , a] = [c∗N , a∗ ] = 1 (G,β,γ,s)
and stabilize W−N
4.6
(G,β,γ,s)⊥
(and WN
(4.118)
).
The pair correlation function in the hypergeometric case
Specializing the Christoffel-Darboux expresssion (3.86) for the pair correlation function ˜ 2g (z, w) to the case (4.23), it may be expressed as follows K Proposition 4.4. ˜ 2(G,β,γ,s) (z, w) K
P∞ P∞ ˜(G,β,s) ∗ w−j (z)w−i (w) − w1 (z)w1∗ (w) j=0 Aij i=0 = z−w
where
i+1 X
(G,β,s) A˜ij :=
(4.119)
rkG (β)hi−k+1 (−β −1 s)hj+k+1 (β −1 s).
(4.120)
k=−j−1 −1 Here we have used the fact that, for g = Cρ γ− (β −1 s),g00 g−1,−1 = γ. ± Alternatively, in terms of the notation {Ψj }j∈N
˜ 2(G,β,γ,s) K
1 1 , x x0
xx0 γ −1
=
P ∞
− 0 ˜(G,β,s) Ψ+ (x)Ψ− (x0 ) − Ψ+ A (x)Ψ (x ) 0 0 j+1 i+1 i,j=0 ij x − x0 37
.
(4.121)
Equivalently, for 0
0 −1
K(x, x ) := −γ(xx ) we have
P∞ 0
K(x, x ) =
i,j=0
˜ 2(G,β,γ,s) K
1 1 , x x0
,
− 0 Aij Ψ+ i (x)Ψj (x ) , x − x0
(4.122)
(4.123)
where (G,β,s) Aij = −A˜j−1,i−1 = −
j X
(G,β)
rk
hj−k (−β −1 s)hi+k (β −1 s). i, j = 1, 2, . . . ,
(4.124)
k=−i
A00 = 1,
4.7
A0j = Ai0 = 0.
Finiteness and generating function for Christoffel-Darboux matrix
In this section we prove that, for polynomial G(z) and S(z), the numerator of the correlator (4.119), viewed as an integral operator kernel into W , is of finite rank, and that in fact the expression may be written as a finite sum. Define the following power series in two variables (r, t) d 1 d (4.125) − tG S(r) + βr A(r, t) := r G S(t) − βt dt dr r−t Proposition 4.5. The generating function for the Christoffel-Darboux coefficients Aij defined by (4.124) is given by ∞ X ∞ X A(r, t) = Aij ri tj . (4.126) i=0 j=0
In particular, if si = 0, ∀ i > L and gi = 0, ∀ i > M , then Aij = 0 ∀ i + j > LM.
(4.127)
To prove this, we begin with the following easily proved generalization of the orthogonality relations satisfied by the complete symmetric functions, valid in the case when only the first L variables (s1 , . . . , sL , 0, 0, · · · ) are nonzero. Lemma 4.6. ξ(s,x)
e
D e
−ξ(s,x)
e
k −ξ(s,x)
k ξ(s,x)
D e
=
=
∞ X N =0 ∞ X
x
N
N X
nk hn (−s)hN −n (s)
(4.128)
nk hn (s)hN −n (−s)
(4.129)
n=0
x
N =0
38
N
N X n=0
Moreover, if si = 0, ∀ i > L, the following orthogonality relations are identically satisfied N X
nk hn (−s)hN −n (s) = 0,
∀N > kL.
(4.130)
n=1
Proof. For each N ≥ 0, define the following generating series S(x, y) := eξ(s,y)−ξ(s,x) ∞ X =: SN (x, y)
(4.131) (4.132)
N =0
where the SN (x, y) are polynomials of degree N in the variables (x, y) SN (x, y) :=
N X
xn y N −n hn (−s)hN −n (s).
(4.133)
n=0
For all k ∈ N, we also define two weighted versions of the polynomials SN (x, y) SN,k (x, y) :=
N X
nk xn y N −n hn (−s)hN −n (s)
(4.134)
n=0
Denoting the Euler operator in x d dx and applying its kth power to S(x, y), we obtain D := x
k
D S(x, y) = P (x)S(x, y) =
(4.135)
∞ X
SN,k (x, y),
(4.136)
N =0
where P (x) is the series P (x) = eξ(s,x) Dk e−ξ(s,x) .
(4.137)
For x = y we have P (x) =
∞ X N =0
x
N
N X
nk hn (−s)hN −n (s)
(4.138)
n=0
which proves the first equation in 4.129. The second equation follows from the substitution s → −s. If si = 0, ∀ i > L, then P (x) and is a polynomial of degree kL. In this case (4.138) implies (4.130). We now turn to the proof of Proposition 4.5. 39
Proof. It is obvious that it is sufficient to prove it for β = 1. For (4.125) we have ξ(s,r)−ξ(s,t) d d e ξ(s,t)−ξ(s,r) r G −t − tG r (4.139) A(r, t) = e dt dr r−t ∞ m X d t d t ξ(s,t)−ξ(s,r) ξ(s,r)−ξ(s,t) G −t = e −G r e (4.140) dt dr r r m=0 ∞ m X t d t −ξ(s,r) d ξ(s,t) −ξ(s,t) ξ(s,r) . = e G −t − m e − e G r −m−1 e r dt r dr m=0 (4.141) Then, from (4.137) and (4.138) we conclude ∞ m X ∞ X N X t
A(r, t) =
m=0
=
∞ X N X
r
c(N, n)
N =0 n=0
tN G(−n − m) − rN −1 tG(N − n − m − 1) c(N, n) (4.142)
N =0 n=0 ∞ X
tm+N r−m G(−n − m) − tm+1 rN −m−1 G(N − n − m − 1) , (4.143)
m=0
where we introduced c(N, n) := hn (−s)hN −n (s).
(4.144)
In the first term we can change the summation variable m 7→ m + 1 − N to get A(r, t) = 1 +
∞ X N X
c(N, n)
−
c(N, n)
N =1 n=0 N ∞ X X
= 1−
tm+1 rN −m−1 G(N − n − m − 1)
(4.145)
m=N −1
N =1 n=0 ∞ X N X
∞ X
∞ X
tm+1 rN −m−1 G(N − n − m − 1)
(4.146)
m=0
c(N, n)
N −2 X
tm+1 rN −m−1 G(N − n − m − 1),
(4.147)
m=0
N =2 n=0
so that indeed for i, j > 0 Aij = −
i+j X
G(j − n)hn (−s)hi+j−n (s).
(4.148)
n=0
If M and L are finite, then (4.125) is a polynomial of total degree LM , and (4.127) follows. For arbitrary β we have Aij = −
i+j X
G(β(j − n))hn (−β −1 s)hi+j−n (β −1 s).
n=0
40
(4.149)
which coincides with eq (4.124). In terms of A˜ eq (4.127) is equivalent to (G,β,s) A˜ij = 0 if i ≥ LM
or j ≥ LM.
(4.150)
and therefore ˜ 2(G,β,γ,s) (z, w) K
PLM −1 PLM −1 ˜(G,β,s) ∗ (w) − w1 (z)w1∗ (w) Aij w−j (z)w−i i=0 j=0 . = z−w
Equivalently, we have the finite form of eq. (4.123) PLM PLM + − 0 i=0 j=0 Aij Ψi (x)Ψj (x ) 0 K(x, x ) = . x − x0
4.8
(4.151)
(4.152)
The multicurrent correlators as generating functions for weighted Hurwitz numbers
In this section, we specialize the multicurrent correlators (3.90) to the case of of hypergeometric τ -functions, for which gˆ = Cˆρ γˆ− (β −1 s). (4.153) In this case Jn (x1 , . . . , xn ) := h0|
n Y
! J+ (xi ) Cˆρ γˆ− (β −1 s)|0i.
(4.154)
i=1
The correlators Wn are therefore expressed fermionically as 1
Wn (x1 , . . . , xn ) = Qn
i=1 xi
h0|
n Y
J+ (xi )Cˆρ γˆ− (β −1 s)|0i.
(4.155)
i=1
We now introduce another set of generating functions for weighted Hurwitz numbers, both for the connected and nonconnected case. Definition 4.4. Fn (s; x1 , . . . , xn ) := F˜n (s; x1 , . . . , xn ) := =
X
X
µ,ν, `(µ)=n
d
X
X
µ,ν, `(µ)=n
d
X
γ |µ| β d−`(ν) HGd (µ, ν) | aut(µ)|mµ (x1 , . . . , xn )pν (s) (4.156) ˜ Gd (µ, ν) | aut(µ)|mµ (x1 , . . . , xn )pν (s) γ |µ| β d−`(ν) H (4.157)
β n+2g−2 F˜g,n ,
(4.158)
g
F˜g,n (s; x1 , . . . , xn ) :=
X
n+`(ν)+2g−2
˜ γ |µ| H G
µ,ν, `(µ)=n
41
(µ, ν) | aut(µ)|mµ (x1 , . . . , xn )pν (s) ,
(4.159) where mµ (x1 , . . . , xn ) denotes the monomial sum symmetric function corresponding to the partition µ, and (g, d) are related by the Riemann-Hurwitz formula 2 − 2g = `(µ) + `(ν) − d.
(4.160)
Note that Fn , F˜n belong to K[x1 , . . . , xn ; s; β, β −1 ][[γ]] and F˜g,n to K[x1 , . . . , xn ; s][[γ]]. ˜ g,n (s; x1 , . . . , xn ) for fixed genus g We may similarly define the multicurrent correlator W and n as the coefficients in the expansion X ˜ n (s; x1 , . . . , xn ) =: ˜ g,n (s; x1 , . . . , xn ). W β n+2g−2 W (4.161) g
The following result, relating the generating functions Fn , F˜n , F˜g,n to the multicurrent correlators, is proved in [1]. Proposition 4.7. ∂n Fn (s; x1 , . . . , xn ), ∂x1 · · · ∂xn ∂n ˜ n (s; x1 , . . . , xn ) = W F˜n (s; x1 , . . . , xn ), ∂x1 · · · ∂xn ∂n ˜ Wg,n (s; x1 , . . . , xn ) = F˜g,n (s; x1 , . . . , xn ). ∂x1 · · · ∂xn Wn (s; x1 , . . . , xn ) =
5
(4.162) (4.163) (4.164)
Bose-Fermi correspondence and “cut and join” operators
The cut-and-join description of the single and double Hurwitz numbers was introduced in [18, 49]. It is known that generating functions of general weighted Hurwitz numbers (or, equivalently, hypergeometric τ -functions) can be described by several versions of cutand-join equations, see e.g. [5–7, 29, 32, 43, 46]. Here we describe two different cut-and-join representations using the bosonization procedure to express these relations both bosonically and in terms of fermionic VEV’s.
5.1
Bose-Fermi correspondence and cut and join representation of the τ -function
One of the cut-and-join type representations is a direct generalization of [18, 49] and is naturally parametrized by the coefficients of log(G) ∞ X (−1)k+1 β k Ak xk := log(γG(βx)). k=0
42
(5.1)
introduced in Section 4.2. From (4.36) it follows that element (4.25) can be represented as Aˆ = log γ
∞ X
j:
ψj ψj∗
:+
j=−∞
=
∞ X
∞ X
(−1)k+1 β k Ak resz=0 (: ψ(z)pk (−D − 1) ψ ∗ (z) :) (5.2)
k=1
ˆk, (−1)k+1 β k Ak Q
(5.3)
k=0
where the operators ˆ k := resx=o Q =
∞ X
∂ ∗ − 1 ψ (x) : : ψ(x)pk −x ∂x
(5.4)
pk (i):ψi ψi∗ :
(5.5)
i=−∞
commute with each other
h
i ˆk, Q ˆ m = 0, Q
k, m ≥ 0.
(5.6)
From (4.25) and (5.3), we may express the exponents Ti in terms of the Ai ’s as ∞ X
Ti =
(−1)k+1 β k Ak pk (i),
i ∈ Z.
(5.7)
k=0
The bose-fermi correspondence is defined by the bosonization map M0 : F0 → B0 M0 |vi 7→ h0|ˆ γ+ (t)|vi
(5.8)
from the (zero charge sector of the) Fermi Fock space F0 ⊂ F to the corresponding sector B0 of the Bosonic Fock space B0 . The latter is viewed as consisting of symmetric functions of an infinite number of auxiliary Bosonic variables {ξa }a∈N+ , in terms of which the KP flow parameters t = (t1 , t2 , . . . ) are understood as normalized power sums ∞
1X i ti := ξ , i a=1 a
i = 1, . . . .
(5.9)
This gives rise to the identity h0|ˆ γ+ (t) : ψ(z)ψ ∗ (w) : |vi =
1 z−w
∗ ∗
eφ(z)−φ(w) ∗∗ − 1 h0|ˆ γ+ (t)|vi,
∀ |vi ∈ F0 ,
(5.10)
where ∗∗ . . . ∗∗ denotes the bosonic normal ordering, which puts all derivatives with respect to tk to the right of all multiplications by tk ’s, and φ is the bosonic operator ∞ X 1 ∂ k φ(x) := tk x − k . (5.11) kx ∂tk k=1 43
This allows us to construct a bosonic counterpart for any bilinear fermionic combination aij : ψi ψj∗ :. In particular, for the operators (5.4), we have the bosonic representation Qk , which are polynomials in tk and ∂t∂k , such that
P
ˆ k = Qk h0|ˆ h0|ˆ γ+ (t)Q γ+ (t).
(5.12)
From (5.10) it immediately follows [3, 43] that a generating function of the bosonic counterparts of these operators Qk is ∞ X ak k=0
1 −1 ∗ φ(xea/2 )−φ(xe−a/2 ) ∗ Qk = a/2 res x e −1 . ∗ ∗ k! (e − e−a/2 )2 x=0
(5.13)
The Qk ’s also generate a commutative algebra [Qk , Qm ] = 0,
k, m ≥ 0.
(5.14)
For example,
Q1 Q2
∞ X
∂ , ∂t k k=1 1X ∂ ∂2 = abta tb + (a + b)ta+b , 2 a,b ∂ta+b ∂ta ∂tb ∞ 1 X ∂3 ∂ = abc ta tb tc + (a + b + c) ta+b+c 3 a,b,c=1 ∂ta+b+c ∂ta ∂tb ∂tc
Q0 =
+
ktk
∞ a+b−1 ∞ 1 X X 1X ∂2 ∂ c(a + b − c) tc ta+b−c + a(a2 − 1) ta . 2 a,b=1 c=1 ∂ta ∂tb 6 a=1 ∂ta
(5.15) (5.16)
(5.17)
From (5.12) we have ∞ X ˆ h0|ˆ γ+ (t)eA = exp (−1)k+1 β k Ak Qk
! h0|ˆ γ+ (t).
(5.18)
k=0
This gives a cut-and-join type representation for the τ -function (1.26): ˆ
τ (G,β,γ) (t, s) = h0|ˆ γ+ (t)eA γˆ− (s)|0i ! ∞ X k+1 k = exp (−1) β Ak Qk h0|ˆ γ+ (t)ˆ γ− (s)|0i = exp
k=0 ∞ X
! (−1)k+1 β k Ak Qk
k=0
exp
∞ X k=1
44
(5.19) (5.20)
! ktk sk .
(5.21)
For the case of simple Hurwitz numbers, considered in Section 6.1 when G(z) = exp(z), if we put γ = 1, (5.21) reduces to P∞
τ (t, s) = eβQ2 e(
k=1
ktk sk )
,
(5.22)
which coincides with the cut-and-join representation of [18, 49] for simple (double) Hurwitz numbers.
5.2
Cut and join equations
If the Ak ’s are viewed as the independent parameters defining the weighting, the τ -function τ (G,β,γ) (t, s) satisfies the generalized cut-and-join equation ∞
X ∂ ∂ (G,β,γ) β τ (G,β,γ) (t, s) = kAk τ (t, s). ∂β ∂A k k=1
(5.23)
It follows from (5.12) and the commutation relations (5.14) that τ (G,β,γ) (t, s) also satisfies an infinite number of further cut-and-join equations corresponding to the variables Ak and γ ∂ (G,β,γ) τ (t, s) = −Q0 τ (G,β,γ) (t, s), ∂γ ∂ (G,β,γ) τ (t, s) = (−1)k+1 β k Qk τ (G,β,γ) (t, s). ∂Ak
−γ
(5.24) (5.25)
For the case where G(z) is a polynomial, we have an alternative cut-and-join representation for the single Hurwitz numbers [29, 50]. Namely, ˆ A J
ˆ
eA eJ−1 |0i = ee
ˆ −A −1 e
|0i.
(5.26)
ˆ ˆ ˆ ˆ eA J−1 e−A = eA res z −1 ψ(z)ψ ∗ (z) e−A
(5.27)
Here z=0 −1
= res z (eT (D) ψ(z))e−T (−D−1) ψ ∗ (z)) z=0 = res (e−T (D) z −1 eT (D) ψ(z))ψ ∗ (z)
(5.28)
= res ((R∗ ψ(z))ψ ∗ (z))
(5.30)
(5.29)
z=0
z=0
= γJ−1 + γ
M X
gk β k res z −1 (Dk ψ(z))ψ ∗ (z) ,
k=1
z=0
(5.31)
where we have used (4.48). Thus τ
(G,β,γ)
(t, s = δk,1 ) = exp γ
t1 +
M X k=1
45
!! k
gk β Vk
·1
(5.32)
where Vk are the bosonic operators corresponding to res z −1 (Dk ψ(z))ψ ∗ (z) . For example z=0
V1 = V2 =
∞ X
ktk
k=1 ∞ X
∂ , ∂tk−1
ktk ltl
k,l=1
6
(5.33) ∂
∂tk+l−1
∂2 + (k + l + 1)tk+l+1 ∂tk ∂tl
.
(5.34)
Examples
To conclude, we focus upon four specific examples of weighted Hurwtz numbers: the simple (single and double) Hurwitz numbers of [35, 39]; the strongly monotonic Hurwitz numbers, corresponding to weights supported curves with theree branch points (Belyi curves) [9,22,29]; the weakly monotonic Hurwitz numbers, corresponding to signed weightings [16, 20, 22] and the quantum Hurwitz numbers introduced in [20, 22]. Explicit formulae are provided for: the content product coefficients entering in the τ -function series, the parameters defining their fermionic respresentation, the weighting factors defining the Hurwitz numbers, and the classical and quantum spectral curves entering in the topological recursion approach.
6.1
Simple (single and double) Hurwitz numbers [35, 39]
This is the original case studied by Pandharipande and Okounkov. It corresponds to the exponential weight generating function G(z) = Exp(z) = ez .
(6.1)
The content product formula entering in the Schur function series (1.26), (1.27) for the τ -function is β P`(λ) (Exp,β) rλ = e 2 i=1 λi (λi −2i+1) . (6.2) The weight WExp (µ(1) , . . . , µ(k) ) is the Dirac measure (i.e. the characteristic function) supported on the partitions consisting only of 2-cycles µ(i) = (2, (1)N −2 ), (1)
(k)
WExp (µ , . . . , µ ) =
i = 1, . . . , k, k Y
δ(µ(i) ,(s,(1)N −2 )
(6.3) (6.4)
i=1
and the exponents Ti (β, γ) entering in (4.8), (4.10) and (4.23) are Ti (β, γ) = iln(γ) + 46
β i(i + 1). 2
(6.5)
The quantum spectral curve is [5]: βDΨ+ 0 (x) −
X
isi γ i xi eiβD Ψ+ 0 (x) = 0
(6.6)
i d where D = x dx is the Euler operator, and the classical spectral curve is X y= isi γ i xi−1 eixy .
(6.7)
i
6.2
Three branch points (Belyi curves): strongly monotonic paths in the Cayley graph [9, 22, 29]
For this case, the weight generating function is linear G(z) = 1 + z =: E(z).
(6.8)
The content product formula is (E,β)
rλ
= β |λ| (1/β)λ ,
(6.9)
where `(λ) Y uλ := (u − i + 1)λi i=1
(u)j := u(u + 1) · · · (u + j − 1)
(6.10)
is the multiple Pochhammer symbol corresponding to the partition λ. The measure WE (µ(1) , . . . , µ(k) ) is supported on the set of k = 1 partitions, and hence corresponds to enumeration of coverings with just three branch points (µ(1) , µ, ν) (Belyi curves). WE (µ(1) , . . . , µ(k) ) =
δk,1 . ∗ ` (µ(1) )
(6.11)
The exponents Ti (β) entering in (4.8), (4.10) and (4.23) are TiE (β, γ)
= iln(γ) +
i X
E T−i (β)
ln(1 + jβ),
= −iln(γ) −
j=1
i−1 X
ln(1 − jβ),
i>0
(6.12)
j=1
The quantum spectral curve is [5]: X βDΨ+ (x) − isi γ i xi (1 + βD)i Ψ+ 0 (x) = 0 0
(6.13)
i
and the classical one is xy =
X
isi γ i (x(1 + xy))i .
i
47
(6.14)
6.3
Signed Hurwitz numbers: weakly monotonic paths in the Cayley graph [16, 20, 22]
In this case the weight generating function is 1 =: H(z). 1−z
˜ G(z) =
(6.15)
The content product coefficient is (H,β)
rλ
= (−β)−|λ| ((−1/β))−1 λ .
(6.16)
The measure WH (µ(1) , . . . , µ(k) ) is thus given by the evaluation of the “forgotten” symmetric functions fλ (c) for the partition with parts {λi } = {`∗ (µ(i) )} at c = (1, 0, . . . ) WH (µ(1) , . . . , µ(k) ) = (−1)d+k Qk
k!
(6.17)
i=1 mi (λ)!
where mi (λ) is the number of colengths `∗ (µ(j) ) equal to i. The exponents Ti (β) entering in (4.8), (4.10) and (4.23) are TiH (β, γ)
:= iln(γ) −
i X
H T−i (β)
ln(1 − jβ),
= −iln(γ) +
j=1
i−1 X
ln(1 + jβ),
i>0
(6.18)
j=1
The quantum spectral curve is [5]: βDΨ+ 0 (x) −
X
isi γ i xi (1 − βD)−i Ψ+ 0 (x) = 0
(6.19)
i
and the classical one is xy =
X
isi γ i xi (1 − xy)−i .
(6.20)
i
6.4
Simple quantum Hurwitz numbers [20, 22]
Choosing a real parameter q, 0 ≤ q ≤ 1, we select the parameters ci to be ci = q i ,
i = 1, 2, . . . .
(6.21)
The weight generating function for this case is therefore ∞ Y G(z) = (1 + q i z) =: E 0 (q) = i=1
48
1 1 −( 1−q )Li2 (q,−z) , e 1+z
(6.22)
where Li2 (q, z) is the quantum dilogarithm function Li2 (q, z) = (1 − q)
∞ X i=1
zi . i(1 − q i )
(6.23)
This gives the “simple quantum” Hurwitz numbers introduced in [20], [22]. The content product coefficient for this case is (E 0 (q),β) rj
(E 0 (q),β)
rλ
=
(z) = =
∞ Y
(1 + q k βj) = (−qβj; q)∞ ,
k=1 ∞ Y
Y
(1 + q k β(j − i)) =
k=1 (i,j)∈λ ∞ Y k |λ|
(6.24) Y
(−qβ(j − i); q)∞
(i,j)∈λ
(βq ) (1/(βq k ))λ ,
(6.25)
k=1
The weight WE 0 (q) (µ(1) , . . . , µ(k) ) for this case is WE 0 (q) (µ(1) , . . . , µ(k) ) := mλ (q, q q , . . . ) ∗ (σ(1)) ) ∗ (σ(k)) ) X q k` (µ · · · q ` (µ 1 = | aut(λ)| σ∈S (1 − q `∗ (µ(σ(1)) ) ) · · · (1 − q `∗ (µ(σ(1)) · · · q `∗ (µ(σ(k)) ) ) k X 1 1 = , (σ(1)) ∗ ∗ ) − 1) · · · (q −` (µ(σ(1)) ) · · · q −`∗ (µ(σ(k)) ) − 1) | aut(λ)| σ∈S (q −` (µ k (6.26) which may be viewed as the (unnormalized) probability measure of a Bosonic gas with energy levels E(µ) := `∗ (µ)~ω0 (6.27) associated to branch points with ramification profiles µ, where q = e−β~ω0 .
(6.28)
The exponents Ti (β) entering in (4.8), (4.10) and (4.23) are E 0 (q) Ti (β, γ)
:= iln(γ) +
i X ∞ X
ln(1 + kβq j )
(6.29)
k=1 j=1 E 0 (q)
T−i
(β, γ) = −iln(γ) −
i−1 X ∞ X
ln(1 − kβq j ) i > 0
(6.30)
k=1 j=1
The quantum spectral curve is: βDΨ+ 0 (x) −
X i
isi γ i xi
∞ Y j=1
49
!i 1 − q j βD
Ψ+ 0 (x) = 0
(6.31)
and the classical one is xy =
X
i i
isi γ x (
i
Appendix A
∞ Y
1 − q j xy)i .
(6.32)
j=1
Fermionic representation of multipair correlators
In the following, we assume that g is lower triangular. This implies, in particular, that the left vacuum is stabilized by both gˆ and gˆ−1 h0|ˆ g = h0|ˆ g −1 = h0|.
A.1
(A.1)
˜ g (z, w) Fermionic representation of K 2
We provide here and in Appendix A.2 the details of the proofs of Proposition 3.7, Lemma ˜ 2g (z, w) may be 3.9 and Corollaries 3.10, 3.11. As defined in eq. (3.77), the pair correlator K expressed fermionically as ( g −1 ψ(z)ψ ∗ (w)ˆ g |0i if |z| > |w| ˜ 2g (z, w) = h0|ˆ (A.2) K −1 ∗ −h0|ˆ g ψ (w)ψ(z)ˆ g |0i if |z| < |w|. We now prove Proposition 3.7, which states that this is equivalent to −1 −1 ˜ 2g (z, w) = τg ([w ] − [z ]) . K z−w
(A.3)
The general case will then follow from Wick’s theorem, as shown in the next subsection. Proof. We begin by proving the stronger relations, for all partitions λ, ( h0|ψ(z)ψ ∗ (w)|λi, if |z| > |w|, sλ ([w−1 ] − [z −1 ]) h0|ˆ γ+ ([w−1 ] − [z −1 ])|λi = = (A.4) z−w z−w −h0|ψ ∗ (w)ψ(z)|λi, if |z| < |w|, which, since these are valid for all λ, implies ( h0|ψ(z)ψ ∗ (w), if |z| > |w|, h0|ˆ γ+ ([w−1 ] − [z −1 ]) = z−w −h0|ψ ∗ (w)ψ(z), if |z| < |w|,
(A.5)
First, note that the LHS of (A.4) vanishes unless λ is either the trivial partitition λ = ∅ , in which case s∅ ([w−1 ] − [z −1 ]) = 1, (A.6) 50
or a hook partition λ = (a|b), with arm length a ≥ 0 and leg length b ≥ 0, for which s(a|b) ([w−1 ] − [z −1 ]) = (−1)b (z − w)z −b−1 w−a−1 .
(A.7)
Turning to the RHS, by Wick’s theorem, h0|ψ(z)ψ ∗ (w)|λi and h0|ψ ∗ (w)ψ(z)|λi also vanish unless λ = ∅ or λ = (a|b). For the first case, if |z| > |w| XX h0|ψ(z)ψ ∗ (w)|0i = z i w−j−1 h0|ψi ψj∗ |0i i∈Z j∈Z ∞ X
1 = w
i=1
w i 1 , = z z−w
(A.8)
and similarly, if |z| < |w| h0|ψ ∗ (w)ψ(z)|0i = =
XX
z i w−j−1 h0|ψj∗ ψi |0i
i∈Z j∈Z ∞ X
1 w
i=0
z i 1 = . w w−z
(A.9)
For λ = (a|b), and |z| > |w| h0|ψ(z)ψ ∗ (w)|(a|b)i =
XX
∗ z i w−j−1 h0|ψi ψj∗ (−1)b ψa ψ−b−1 |0i
i∈Z j∈Z b −a−1 −b−1
= (−1) w
z
(A.10)
by Wick’s theorem, while for |z| < |w|, XX ∗ h0|ψ ∗ (w)ψ(z)|(a|b)i = z i w−j−1 h0|ψj∗ ψi (−1)b ψa ψ−b−1 |0i i∈Z j∈Z
= −(−1)b w−a−1 z −b−1 .
(A.11)
Taking the scalar product of (A.5) with gˆ|0i then proves the equivalence of eq. (A.2) with (A.3).
A.2
˜ g (z, w) The n-pair correlator K 2n
For a set of n nonzero complex numbers z = (z1 , z2 , . . . zn ), we use the notations [z] and [z−1 ] to denote the infinite sequence {[z]i }i=1,2,··· , of normalized power sums n
1X [z]i := (zj )i i j=1
n
1 X −1 i and [z ]i := (z ) i j=1 j −1
We now proceed to the proof of Lemma 3.9, which is: 51
(A.12)
Lemma A.1. For 2n distinct nonzero complex parameters {zi , wi }i=1,...n , we have n Y 1 ∗ h0| (ψ(zi )ψ (wi )) = det h0|ˆ γ+ ([w−1 ] − [z−1 ]), z − w i j 1≤i,j≤n i=1
(A.13)
and Corollaries 3.10, 3.11. Proof. We proceed by induction on n, recalling that the Cauchy determinant is given by n(n−1) ∆(z)∆(w) 1 = (−1) 2 Q (A.14) det zi − wj 1≤i,j,≤n (zi − wj ) where ∆(z) =
Y
(zi − zj ),
∆(w) =
1≤i |w|, inserting a sum over a complete set of intermediate states, we have X h0|ˆ g −1 ψ(z)ψ ∗ (w)ˆ g |0i = h0|ˆ g −1 ψ(z)ˆ g |λ; −1ihλ; −1|ˆ g −1 ψ ∗ (w)ˆ g |0i (A.25) λ
By Wick’s theorem, the only partitions that contribute to this sum are those of the form λ = (1)j , j = 0, 1, . . . . Substituting ∗ |(1)j ; −1i = (−1)j ψ−j−1 |0i,
53
(A.26)
into the sum therefore gives −1
∗
h0|ˆ g ψ(z)ψ (w)ˆ g |0i =
∞ X
∗ h0|ˆ g −1 ψ(z)ˆ g ψ−j−1 |0ih0|ψ−j−1 gˆ−1 ψ ∗ (w)ˆ g |0i
j=0
=
∞ X
g∗ wjg (w)w−j+1 (z).
(A.27)
j=1
A similar calculation shows, for |z| < |w| −1
∗
h0|ˆ g ψ (w)ψ(z)ˆ g |0i =
∞ X
h0|ˆ g −1 ψ ∗ (w)ˆ g ψj |0ih0|ψj∗ gˆ−1 ψ(z)ˆ g |0i
j=0
=
∞ X
g w−j+1 (w)wjg∗ (z).
(A.28)
j=1
Acknowledgements. The work of A.A. is supported by IBS-R003-D1 and by RFBR grant 15-0104217. B.E. wishes to thank the Centre de recherches math´ematiques, Montr´eal, for the Aisenstadt Chair grant, and the FQRNT grant from the Qu´ebec government, that partially supported this joint project. G.C. acknowledges support from the European Research Council, grant ERC-2016STG 716083 “CombiTop”, from the Agence Nationale de la Recherche, grant ANR 12-JS02-001-01 ´ “Cartaplus” and from the City of Paris, grant “Emergences 2013, Combinatoire `a Paris”. The work of J.H. was partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Fonds de recherche du Qu´ebec, Nature et technologies (FRQNT). ´ He wishes to thank the Institut des Hautes Etudes Scientifiques, where much of this work was completed. It was also supported by the ERC Starting Grant no. 335739 “Quantum fields and knot homologies” funded by the European Research Council under the European Union’s Seventh Framework Programme. The authors would also like to thank the organizers of the January March, 2017 thematic semester “Combinatorics and interactions” at the Institut Henri Poincar´e, where they were participants during the completion of this work.
References [1] A. Alexandrov, G. Chapuy, B. Eynard and J. Harnad “Weighted Hurwitz numbers and topological recursion: an overview”, arXiv:1610.09408 [2] A. Alexandrov, G. Chapuy, B. Eynard and J. Harnad “Weighted Hurwitz numbers and topological recursion”, preprint CRM-3359 (2017). [3] A. Alexandrov, “Matrix Models for Random Partitions,” Nucl. Phys. B 851 620 (2011). [4] A. Alexandrov, “Enumerative Geometry, Tau-Functions and Heisenberg-Virasoro Algebra,” Commun. Math. Phys. 338, no. 1, 195 (2015).
54
[5] A. Alexandrov, D. Lewanski and S. Shadrin, “Ramifications of Hurwitz theory, KP integrability and quantum curves,” JHEP 1605 124 (2016). [6] A. Alexandrov, A. Mironov, A. Morozov and S. Natanzon, “Integrability of Hurwitz Partition Functions. I. Summary,” J. Phys. A 45 045209 (2012). [7] A. Alexandrov, A. Mironov, A. Morozov and S. Natanzon, “ On KP-integrable Hurwitz functions”, JHEP 1411 080 (2014). [8] A. Alexandrov and A. Zabrodin, “Free fermions and tau-functions,” J. Geom. Phys. 67, 37 (2013). [9] J. Ambjørn and L. Chekhov, “The matrix model for dessins d’enfants”, Ann. Inst. Henri Poincar´e, Comb. Phys. Interact. 1, 337-361 (2014). [10] G. Borot, B. Eynard, M. Mulase and B. Safnuk, “A matrix model for Hurwitz numbers and topological recursion”, J. Geom. Phys. 61, 522–540 (2011). [11] G. Borot and B Eynard, “All-order asymptotics of hyperbolic knot invariants from nonperturbative topological recursion of A-polynomials”, Quantum Topology 6, 39-138 (2015). [12] B. Eynard and N. Orantin, “Topological recursion in enumerative geometry and random matrices”, J. Phys. A 42 293001 (2009). [13] B. Eynard, and N. Orantin, “Invariants of spectral curves and intersection theory of moduli spaces of complex curves”, Commun. Num. Theor. Phys. 8 541-588 (2014) ¨ [14] G. Frobenius, “Uber die Charaktere der symmetrischen Gruppe”, Sitzber. Akad. Wiss., Berlin, 516-534 (1900). Gesammelte Abhandlung III, 148-166. ¨ [15] G. Frobenius, “Uber die Charakterische Einheiten der symmetrischen Gruppe”, Sitzber. Akad. Wiss., Berlin, 328-358 (1903). Gesammelte Abhandlung III, 244-274. [16] I. P. Goulden, M. Guay-Paquet and J. Novak, “Monotone Hurwitz numbers and the HCIZ Integral”, Ann. Math. Blaise Pascal 21 71-99 (2014). [17] I. P. Goulden, M. Guay-Paquet and J. Novak, “Toda Equations and Piecewise Polynomiality for Mixed Double Hurwitz numbers”, SIGMA 12, 040 (2016). [18] I. P. Goulden and D. M. Jackson, “Transitive factorisation into transpositions and holomorphic mappings on the sphere”, Proc. Amer. Math. Soc., 125, 51-60. no. 1 (1997). [19] M. Guay-Paquet and J. Harnad, “2D Toda τ -functions as combinatorial generating functions”, Lett. Math. Phys. 105 Page 827-852 (2015). [20] M. Guay-Paquet and J. Harnad, “Generating functions for weighted Hurwitz numbers”, J. Math. Phys. 58, 083503 (2017). [21] J. Harnad, “Quantum Hurwitz numbers and Macdonald polynomials”, J. Math. Phys. 57 113505 (2016).
55
[22] J. Harnad, “Weighted Hurwitz numbers and hypergeometric τ -functions: an overview”, AMS Proc. Symp. P. Math. 93, 289-333 (2016). [23] J. Harnad and A. Yu. Orlov, “Fermionic construction of partition functions for two-matrix models and perturbative Schur function expansions”, J. Phys. A 39 8783–8809 (2006). [24] J. Harnad and A. Yu. Orlov, “Hypergeometric τ -functions, Hurwitz numbers and enumeration of paths”, Commun. Math. Phys. 338, 267-284 (2015). ¨ [25] A. Hurwitz, “Uber Riemann’sche Fl¨asche mit gegebnise Verzweigungspunkten”, Math. Ann. 39, 1-61 (1891). Matematische Werke I, 321-384. ¨ [26] A. Hurwitz, “Uber die Anzahl der Riemann’sche Fl¨asche mit gegebnise Verzweigungspunkten”, Math. Ann. 55, 53-66 (1902). Matematische Werke I, 42-505. [27] M. Jimbo and T. Miwa “Solitons and Infinite Dimensional Lie Algebras”, Publ. RIMS Kyoto Univ. 19, 943–1001 (1983). [28] V. Kac and A. S. Schwarz, “Geometric interpretation of the partition function of 2-D gravity,” Phys. Lett. B 257, 329 (1991). [29] M. Kazarian and P. Zograf, “Virasoro constraints and topological recursion for Grothendieck’s dessin counting”, Lett. Math. Phys. 105 1057–1084 (2015). [30] S. Kharchev, A. Marshakov, A. Mironov and A. Morozov, “Generalized Kazakov-MigdalKontsevich model: Group theory aspects”, Int. J. Mod. Phys. A 10 2015 (1995). [31] I. G. Macdonald, Symmetric Functions and Hall Polynomials, Clarendon Press, Oxford, (1995). [32] A. D. Mironov, A. Yu. Morozov, and S. M. Natanzon, “Complete set of cut-and-join operators in the Hurwitz-Kontsevich theory”, Theor. Math, Phys. 166, 1-22 (2011). [33] M. Mulase, S. Shadrin, and L. Spitz. “The spectral curve and the Schrodinger equation of double Hurwitz numbers and higher spin structures”, Commun. Number Theory and Physics 7, 125–143 (2013). [34] S. M. Natanzon and A. Yu. Orlov, “BKP and projective Hurwitz numbers”, Lett. Math.Phys. 107, 1065–1109 (2017). [35] A. Okounkov, “Toda equations for Hurwitz numbers”, Math. Res. Lett. 7, 447–453 (2000). [36] A. Okounkov and R. Pandharipande, “Gromov-Witten theory, Hurwitz theory, and completed cycles”, Ann. Math. 163, 517-560 (2006). [37] A. Yu. Orlov and D. M. Scherbin, “Hypergeometric solutions of soliton equations”, Theor. Math. Phys. 128, 906-926 (2001). [38] A. Y. Orlov and D. M. Scherbin, “Fermionic representation for basic hypergeometric functions related to Schur polynomials,” Theor. Math. Phys. 137, 1574–1589 (2003).
56
[39] R. Pandharipande, “The Toda Equations and the Gromov-Witten Theory of the Riemann Sphere”, Lett. Math. Phys. 53, 59-74 (2000). [40] M. Sato, “Soliton equations as dynamical systems on infinite dimensional Grassmann manifolds”, RIMS, Kyoto Univ. Kokyuroku 439, 30-46 (1981). [41] M. Sato and Y. Sato, “Soliton equations as dynamical systems on infinite dimensional Grassmann manifold”, in: Nonlinear PDE in Applied Science, Proc. U. S.-Japan Seminar, Tokyo 1982, Kinokuniya, Tokyo, 259-271 (1983). [42] I. Schur “Neue Begr¨ undung’ der Theorie der Gruppencharaktere”, Sitzber. Akad. Wiss., Berlin, 406-432 (1905). [43] S. Shadrin, L. Spitz and D. Zvonkine, “On double Hurwitz numbers with completed cycles,” J. Lond. Math. Soc. 86, 407 (2012). ´ 6, [44] G. Segal and G. Wilson, “Loop groups and equations of KdV type ”, Pub. Math. de l’IHES 5-65 (1985). [45] K. Takasaki, “Initial value problem for the Toda lattice hierarchy”, in: Group Representation and Systems of Differential Equations, Adv. Stud. in Pure Math. 4, 139–163 (1984). [46] K. Takasaki, “Generalized string equations for double Hurwitz numbers”, J. Geom. Phys. 62 1135-1156 (2012). [47] T. Takebe, “Representation theoretical meaning of the initial value problem for the Toda lattice hierarchy I”, Lett. Math. Phys. 21, 77–84 (1991). [48] K. Ueno and K. Takasaki, “Toda Lattice Hierarchy”, in: Group Representation and Systems of Differential Equations, Adv. Stud. in Pure Math. 4, , 1–95 (1984). [49] R. Vakil, Harvard Thesis (1997). [50] P. Zograf, “Enumeration of Grothendieck’s dessins and KP hierarchy”, Int. Math Res. Notices 24, 13533–13544 (2015).
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