Generalized string equations for Hurwitz numbers

Transcription

1 Generalized string equations for Hurwitz numbers Kanehisa Takasaki December 17, Hurwitz numbers of Riemann sphere 2. Generating functions of Hurwitz numbers 3. Fermionic representation of tau functions 4. Generalized string equations 5. Classical limit of generalized string equations 1

2 1. Hurwitz numbers of Riemann sphere The Hurwitz numbers enumerate topologically nonequivalent finite ramified coverings π : Γ Γ 0 of a Riemann surface Γ 0. In the following, we consider the case where Γ 0 = CP 1. Partition as ramification data In a neighborhood of the fiber π 1 (P ) of a point P, Γ looks like a union of cyclic coverings of degree µ 1, µ 2,. They give a partition µ = (µ 1, µ 2, ) = (1 m 1 2 m2 ) of the degree d of the covering. 2

3 1. Hurwitz numbers of Riemann sphere Hurwitz numbers Given a positive integer d, a partition µ (1),, µ (r) of d and r points P 1,, P r of CP 1, we consider coverings π : Γ CP 1 of degree d that are ramified over these points of ramification type µ (1),, µ (r). There are only a finite number of topologically nonequivalent coverings of this type. The Hurwitz number counts the equivalence classes [π] with weight Aut(π): H d (µ (1),, µ (r) ) = [π] 1 Aut(π) 3

4 1. Hurwitz numbers of Riemann sphere Formula (Burnside s theorem) H d (µ (1),, µ (r) ) = λ =d ( ) 2 r dim λ f λ (µ (k) ), d! k=1 dim λ = χ λ (C(1 d )), f λ (µ) = χ λ(c(µ)) dim λ C(µ), where χ λ (C) denotes the irreducible character (class function) of the symmetric group S d for the partition λ, C(µ) the conjugacy class of cycle type µ = (1 m 1 2 m2 ), and C(µ) the cardinality of C(µ) as a subset of S d : C(µ) = d!/z µ, z µ = i 1 m i!i m i. 4

5 2. Generating functions of Hurwitz numbers Generating function of almost simple Hurwitz numbers H d (1 d 2 2,, 1 d 2 2, µ) }{{} r Introduce variables β, Q and x = (x 1, x 2, ), the power sums p k = i 1 xk i and their products p µ = p µ1 p µ2. Define the generating function Z(x) as Z(x) = r=0 d=0 µ =d H d (1 d 2 2,, 1 d 2 2, µ) βr }{{} r! Qd p µ r 5

6 Generating function of double Hurwitz numbers 2. Generating functions of Hurwitz numbers H d (µ, 1 d 2 2,, 1 d 2 2, µ) }{{} r Introduce yet another set of variables x = ( x 1, x 2, ) and their power sums p k = i 1 xk i. Define the generating function Z(x, x) as Z(x, x) = r=0 d=0 µ = µ =d H d (µ, 1 d 2 2,, 1 d 2 2, µ) βr }{{} r! Qd p µ p µ r 6

7 Change of variables for KP and Toda hierarchies 2. Generating functions of Hurwitz numbers t = (t 1, t 2, ) for KP hierarchy: t k = p k k = 1 k i 1 x k i t = (t 1, t 2, ) and t = ( t 1, t 2, ) for Toda hierarchy: t k = p k k = 1 k x k i, i 1 t k = p k k = 1 k Schur functions are redefined as functions of these variables: i 1 s λ (x) = s λ [t], s λ ( x) = s λ [ t] (Zinn-Justin s notation) x k i 7

8 Generating functions in terms of Schur functions Use Frobenius formula µ =d 2. Generating functions of Hurwitz numbers χ λ (C(µ)) p µ = s λ (x) to rewrite the z µ generating functions to sums over partitions λ of arbitrary length: Z(x) = λ dim λ e βκλ/2 Q λ s λ (x) = λ! λ e βκ λ/2 Q λ s λ [t]s λ [1, 0, ], Z(x, x) = λ e βκ λ/2 Q λ s λ (x)s λ ( x) = λ e βκ λ/2 Q λ s λ [t]s λ [ t]. Remark: Relevant formulae dim λ λ! = s λ [1, 0, ], f λ (1 d 2 2) = κ λ = i 1 λ i (λ i 2i + 1). 8

9 2. Generating functions of Hurwitz numbers Cut-and-join operator M 0 = 1 2 k,l=1 ( 2 ) klt k t l + (k + l)t k+l t k+l t k t l Schur functions are eigenfunctions of M 0 : M 0 s λ [t] = κ λ 2 s λ[t]. Z[t] = Z(x) and Z[t, t] = Z(x, x) can be expressed as Z[t] = e βm 0 e Qt 1, ( Z[t, t] = e βm 0 exp ) Q k kt k t k. k=1 9

10 Generating functions as tau functions 2. Generating functions of Hurwitz numbers Z[t] is a tau function of the KP hierarchy (Kazarian & Lando, Goulden & Jackson, ). Z[t, t] is a tau function of the Toda hierarchy at a point, say s = 0, of the lattice (Okounkov). In other words, Z[t, t] is a tau function of the 2-component KP hierarchy. Remark: Any tau function τ(s, t, t) of the Toda hierarchy is a sequence of tau functions (indexed by s Z) of the 2-component KP hierarchy. 10

11 3. Fermionic representation of tau functions Fermionic operators and Fock space creation-annihilation operators ψ i, ψ i (i Z) ψ i ψj + ψj ψ i = δ i+j,0, ψ i ψ j + ψ j ψ i = 0, ψi ψj + ψj ψi = 0 ground states in charge s sector of the Fock space s = ψs 1ψ s, s = ψ s ψ s+1 Fermion bilinears J k = :ψ n+k ψn:, n Z L 0 = n:ψ n ψn:, n Z W 0 = n 2 :ψ n ψn: n Z Remark: M (n 1/2) 2 :ψ n ψn: = 1 2 W L J 0 n Z 11

12 Tau function for double Hurwitz numbers The special GL( ) element g = e βw 0/2 Q L 0 determines a tau function ( ) ( τ(s, t, t) = s exp t k J k e βw0/2 Q L 0 exp k=1 3. Fermionic representation of tau functions ) t k J k s k=1 of the Toda hierarchy. This tau function has the Schur function expansion τ(s, t, t) = e βs(s+1)(2s+1)/12 Q s(s+1)/2 λ e βκ λ/2 (e β(2s+1)/2 Q) λ s λ [t]s λ [ t] and reduces to Z[t, t] upon renormalizing Q and setting s = 0. 12

13 4. Generalized string equations Intertwining relations of fermion bilinears g = e βw 0/2 Q L 0 intertwines special fermion bilinears as J k g = gq k e βk2 /2 n Z e βkn :ψ n+k ψ n:, gj k = Q k e βk2 /2 n Z e βkn :ψ n k ψ n:g Remark: These relations play a role in an integrable structure of the melting crystal model as well. 13

14 4. Generalized string equations Lax and Orlov-Schulman operators M = L = e s + u n e ( n+1) s, L 1 = ū 0 e s + nt n L n + s + Lax equations v n L n, L t k = [B k, L], ū n e (n 1) s, M = n t L n n + s + L t k = [ B k, L], k = 1, 2, same equations replacing L M, Twisted canonical commutation relations L, M v n Ln [L, M] = L, [ L, M] = L 14

15 4. Generalized string equations Theorem The generalized string equations L k = Q k e βk2 /2 Lk e βk M, L k = Q k e βk2 /2 L k e βkm hold for k = 1, 2,. These equations reduce to the lowest ones L = Qe β/2 Le β M, L 1 = Qe β/2 L 1 e βm Remark: Generalized string equations in c = 1 string theory L = L M + const. L, L 1 = L 1 M + const.l 1 15

16 5. Classical limit of generalized string equations Classical (= dispersionless) limit of Toda hierarchy Introduce a new parameter, allow the tau function itself to depend on, and assume that the rescaled tau function τ (s, t, t) = τ(, 1 s, 1 t, 1 t) behaves as log τ (s, t, t) 2 F(s, t, t) + O( 1 ) ( 0). F(s, t, t) is called free energy, etc. In the Lax formalism, this amounts to replacing the difference operators L, M, L, M (quantum observables) by functions L, M, L, M (classical observables): a n (, s)e n s a (0) n (s)p n, a n (, s) = a (0) n (s) + O( ). n n 16

17 Lax and Orlov-Schulman functions 5. Classical limit of generalized string equations M = L = p + nt n L n + s + u (0) n p 1 n, L 1 = ū (0) 0 p 1 + v (0) n L n, M = Lax equations and twisted canonical Poisson relations L t k = {B k, L}, n t n L n + s + L t k = { B k, L}, k = 1, 2, same equations replacing L M, L, M, {L, M} = L, { L, M} = L ū (0) n p n 1, v (0) n L n with respect to Poisson bracket {F, G} = p F p G s F s p G p. 17

18 Classical limit of generalized string equation 5. Classical limit of generalized string equations Rescaling s, t, t as s, t, t 1 s, 1 t, 1 t changes the generalized string equation as L = Qe β/2 Le β 1 M, L 1 = Qe β/2 L 1 e β 1 M These equations themselves do not have a limit as 0. By rescaling β as β 1 β, one can take the classical limit as 0. Thus the generalized string equations L = Q Le β M, L 1 = QL 1 e βm in the classical limit is obtained. 18

19 5. Classical limit of generalized string equations Theorem There is a unique solution of the classical limit of the generalized string equations that has power series expansion with respect to t, t: log ū 0 = log Q + βs + (terms of positive orders in t, t), u n = βn t n ū n 0 + (terms of higher orders in t, t), ū n = βnt n ū 0 + (terms of higher orders in t, t). ( (0) is omitted.) This solution is quasi-homogeneous with respect to rescaling t n c n t n, t n c n t n, s s, p c 1 p, u n c n u n, ū n c n ū n, v n c n v n, v n c n v n. 19

20 5. Classical limit of generalized string equations Solution at special values of t, t When t is specialized to t k = t 1 δ k1 (which amounts to specializing to almost-simple Hurwitz numbers), L simplifies as L = pe β t 1 ū 0 p 1, and ū 0 (= 2 F/ s 2 ) is determined by the single equation log ū 0 = log Q + βs + β k=1 kt k ( kβ t 1 ū 0 ) k k! Remarkably, the functional structure of L essentially coincides with the defining equation z = we w of Lambert s W-function w = W (z) (L 1 z, p 1 w). Presumably, this will be related to the spectral curve in the sense of Eynard and Orantin.. 20