aa r X i v : . [ m a t h - ph ] F e b Weighted Hurwitz numbers, τ -functionsand matrix integrals †∗ J. Harnad , † Department of Mathematics and Statistics, Concordia University1455 de Maisonneuve Blvd. W. Montreal, QC H3G 1M8 Canada Centre de recherches math´ematiques, Universit´e de Montr´eal,C. P. 6128, succ. centre ville, Montr´eal, QC H3C 3J7 Canada
Abstract
The basis elements spanning the Sato Grassmannian element correspond-ing to the KP τ -function that serves as generating function for rationallyweighted Hurwitz numbers are shown to be Meijer G -functions. Using theirMellin-Barnes integral representation the τ -function, evaluated at the traceinvariants of an externally coupled matrix, is expressed as a matrix integral.Using the Mellin-Barnes integral transform of an infinite product of Γ func-tions, a similar matrix integral representation is given for the KP τ -functionthat serves as generating function for quantum weighted Hurwitz numbers. The fact that KP and 2 D -Toda τ -functions of hypergeometric type serve as gener-ating functions for weighted Hurwitz numbers was shown in [2–5], generalizing thecase of simple (single and double) Hurwitz numbers [8, 9]. Sections 1.1 and 1.2below, and Section 2 give a brief review of this theory, together with two illustrativeexamples: rational and quantum weighted Hurwitz numbers. In Section 3, it isshown how evaluation of such τ -functions at the trace invariants of a finite matrixmay be expressed either as a Wronskian determinant or as a matrix integral. Thecontent of subsections 3.2–3.4 are largely drawn from [6, 7], in which further detailsand proofs of the main results may be found. ∗ Text of invited presentation at: Quantum Theory and Symmetries, XIth International sym-posium, Centre de recherches math´ematiques, Montr´eal, July 1-5, 2019. † e-mail: [email protected] .1 Geometric meaning of classical Hurwitz numbers The Hurwitz number H ( µ (1) , . . . , µ ( k ) ) is the number of inequivalent branched N -sheeted covers Γ → P of the Riemann sphere, with k branch points ( Q , . . . , Q k ),whose ramification profiles are given by k partitions ( µ (1) , . . . , µ ( k ) ) of N , normalizedby dividing by the order | aut(Γ) | of its automorphism group. The Euler character-istic χ and genus g of the covering curve is given by the Riemann-Hurwitz formula : χ = 2 − g = 2 N − d, d := l X i =1 ℓ ∗ ( µ ( i ) ) , (1.1)where ℓ ∗ ( µ ) := | µ | − ℓ ( µ ) = N − ℓ ( µ ) is the colength of the partition.The Frobenius-Schur formula gives H ( µ (1) , . . . µ ( k ) ) in terms of S N characters: H ( µ (1) , . . . µ ( k ) ) = X λ, | λ | = N h k − ( λ ) k Y j =1 χ λ ( µ ( i ) ) z µ ( j ) , | µ ( i ) | = N, (1.2)where h ( λ ) = (cid:16) det λ i − i + j )! (cid:17) − is the product of the hook lengths of the partition λ = ( λ ≥ · · · ≥ λ ℓ ( λ > χ λ ( µ ( j ) ) is the irreducible character of representation λ evaluated on the conjugacy class µ ( j ) , and z µ ( j ) := Y i i m i ( µ ( j ) ) ( m i ( µ ( j ) ))! (1.3)is the order of the stabilizer of any element of cyc( µ ( j ) ) (and m i ( µ ( j ) ) = µ ( j ) equal to i ) Define the weight generating function G ( z ), or its dual e G ( z ), as an infinite (or finite)product or sum (formal or convergent). G ( z ) = ∞ Y i =1 (1 + zc i ) = 1 + ∞ X j =1 g j z j e G ( z ) = ∞ Y i =1 (1 − zc i ) − = 1 + ∞ X j =1 e g j z j . (1.4)The weight for a branched covering with ramification profiles ( µ (1) , . . . , µ ( k ) ) is de-fined to be: W G ( µ (1) , . . . , µ ( k ) ) := 1 k ! X σ ∈ S k X ≤ i < ···
Frobenius character formula s λ ( t ) = X µ, | µ | = | λ | χ λ ( µ ) p µ ( t ) z µ , s λ ( s ) = X ν, | ν | = | λ | χ λ ( ν ) p ν ( s ) z ν (2.5)to change the basis of Schur functions to power sum symmetric functions p µ ( t ) := ℓ ( µ ) Y i =1 p µ i ( t ) , p j ( t ) = jt j , p ν ( s ) := ℓ ( ν ) Y i =1 p ν i ( s ) , p j ( s ) = js j . (2.6) Theorem 2.2 (Hypergeometric Toda τ -functions as generating function forweighted double Hurwitz numbers [3, 5]) . The τ -function τ ( G,β ) ( t , s ) can equiva-lently be expressed as a double infinite series in the bases of power sum symmetricfunctions as follows τ ( G,β ) ( t , s ) = ∞ X d =0 X µ,ν, | µ | = | ν | β | µ | + d H dG ( µ, ν ) p µ ( t ) p ν ( s ) . (2.7) It is thus a generating function for the numbers H dG ( µ, ν ) of weighted n -fold branchedcoverings of the sphere, with a pair of specified branch points having ramificationprofiles ( µ, ν ) and genus given by the Riemann-Hurwitz formula − g = ℓ ( µ ) + ℓ ( ν ) − d, d = k X i =1 ℓ ∗ ( µ ( i ) ) . (2.8) Corollary 2.3 (Hypergeometric KP τ -functions as generating functions forweighted single Hurwitz numbers) . Set: s = β − t := ( β − , , , . . . ) .Then the series τ ( G,β ) ( t , β − t ) := τ ( G,β ) ( t ) = X λ ( h ( λ )) − r ( G,β ) λ s λ ( t )= ∞ X d =0 X µ β d H dG ( µ ) p µ ( t ) is a KP τ -function which is a generating function for weighted single numbers H dG ( µ ) for | µ | -fold branched coverings of the sphere, with a branch point having ramificationprofile ( µ ) at Q and genus given by the Riemann-Hurwitz formula. − g = | µ | + ℓ ( µ ) − d. (2.9)5 Wronskian and matrix integral representationof τ ( G,β ) ([ X ]) In [6, 7] new matrix integral representations were derived for the τ -functions thatserve as generating functions for rationally and quantum weighted Hurwitz numbers.The main result is that, using Laurent series and Mellin-Barnes integral representa-tions of the adapted bases for the respective elements of the infinite Grassmanniancorresponding to these cases, the τ -functions may be expressed as Wronskian deter-minants or as matrix integrals. Henceforth, we always set: s = β − t := ( β − , , , . . . ) (3.1)and τ ( G,β ) ( t ) := τ ( G,β ) ( t , β − t ) (3.2)is a KP τ -function of hypergeometric type.For k ∈ Z , define: φ k ( x ) := β πix k − I | ζ | = ǫ ρ ( G,β ) ( ζ ) e β − xζ dζζ k , = βx − k ∞ X j =0 ρ ( G,β ) j − k j ! (cid:18) xβ (cid:19) j , (3.3)where ρ ( G,β ) ( ζ ) := k − X i = −∞ ρ ( G,β ) − i − ζ i . (3.4)Then { φ k (1 /z ) } k ∈ N + is a basis for the element w ( G,β ) of the Sato Grassmannian thatdetermines the KP τ -function τ ( G,β ) ( t ) [1]. Theorem 3.1 (Quantum spectral curve and eigenvalue equations [1]) . The func-tions φ k ( x ) satisfy L φ k ( x ) := ( xG ( β D ) − D ) φ k ( x ) = ( k − φ k ( x ) (3.5) where D := x ddx is the Euler operator. y = G ( βxy ) . (3.6) For G ( z ) = G c , d ( z ), denote φ k ( x ) =: φ ( c , d ,β ) k ( x ). Then ζ L Y l =1 ( D + 1 βc l ) φ ( c , d ,β ) k + ( D + k − M Y m =1 ( D − − βd m ) φ ( c , d ,β ) k = 0 , (3.7)where ζ := − κ c , d x, κ c , d := ( − M Q Ll =1 βc l Q Mm =1 βd m . (3.8) G -functions [6, 7] It may be shown that φ ( c , d ,β ) k has the Mellin-Barnes integral representation: φ ( c , d ,β ) k = C ( c , d ,β ) k G ,LL,M +1 − βc , · · · , − βc L − k, βd , · · · , βd M (cid:12)(cid:12)(cid:12)(cid:12) − κ c , d x ! = C ( c , d ,β ) k πi Z C k Γ(1 − k − s ) Q Lℓ =1 Γ (cid:16) s + βc ℓ (cid:17) ( − κ c , d x ) s Q Mm =1 Γ (cid:16) s − βd m (cid:17) ds. ∼ βρ − k ( c , d )( κx ) k − L F M − k + βc , · · · , − k + βc L − k − βd , · · · , − k − βd M (cid:12)(cid:12)(cid:12)(cid:12) κ c , d x ! (3.9)where C ( c , d ,β ) k := Q Mj =1 Γ( − βd j )( − β ) k − Q Lℓ =1 Γ( βc ℓ ) , (3.10)The contour C k is chosen so that the poles at 1 − k, − k, · · · are to the right andthe poles at {− i − βc j } j =1 , ··· L, i ∈ N + to the left. (See Figure 1.)7 k + 1 − k + 2 · · · · · · N N + 1 − βc ℓ − βc ℓ − · · ·· · · − βc ℓ − βc ℓ − · · ·· · · ℜ s = N + ℜ s = N + C k Figure 1: The contours of integration for the function φ ( c , d ,β ) k in the case L > M + 1.
The following is an integral representation of φ ( H q ,β ) k ( x ), valid for all x ∈ C , φ ( H q ,β ) k = 12 πi Z C k A H q ,k ( s ) x s ds, (3.11)where A H q ,k ( z ) := ( − β ) − k Γ(1 − k − z ) ∞ Y m =0 (cid:18) ( − βq m ) − z Γ( − β − q − m )Γ( z − β − q − m ) (cid:19) . (3.12)The contour C k is defined as starting at + ∞ immediately above the real axis,proceeding to the left above the axis, winding around the poles at the integers s = − k, − k + 1 . . . . in a conterclockwise sense and continuing below the axis backto + ∞ . τ ( G,β ) ( t ) If τ ( G,β ) ( t ) is evaluated at the trace invariants of diagonal X ∈ Mat n × n t = (cid:2) X (cid:3) , t i = 1 i tr X i ,X := diag( x , . . . , x n ) , (3.13)it is expressible as the ratio of n × n determinants τ ( G,β ) (cid:0)(cid:2) X (cid:3)(cid:1) = Q ni =1 x n − i Q ni =1 ρ − i det ( φ i ( x j )) ≤ i,j, ≤ n ∆( x ) , (3.14)8here ∆( x ) = Y ≤ i This work was partially supported by the Natural Sciences andEngineering Research Council of Canada (NSERC) and the Fonds de recherche du Qu´ebec,Nature et technologies (FRQNT). References [1] A. Alexandrov, G. Chapuy, B. Eynard and J. Harnad, “Weighted Hurwitz numbersand topological recursion: an overview”, J. Math. Phys. , 081102: 1-20 (2018).[2] M. Guay-Paquet and J. Harnad, “2D Toda τ -functions as combinatorial generatingfunctions”, Lett. Math. Phys. τ -functions, Hurwitz numbers andenumeration of paths”, Commun. Math. Phys. , 267-284 (2015).[4] J. Harnad, “Weighted Hurwitz numbers and hypergeometric τ -functions: anoverview”, AMS Proceedings of Symposia in Pure Mathematics , 289-333 (2016).[5] M. Guay-Paquet and J. Harnad, “Generating functions for weighted Hurwitz num-bers”, J. Math. Phys. , 083503 (2017).[6] M. Bertola and J. Harnad, “Rationally weighted Hurwitz numbers, Meijer G -functions and matrix integrals”, J. Math. Phys. , 103504 (2019).[7] J. Harnad and B. Runov, “Matrix model generating function for quantum weightedHurwitz numbers”, J. Phys. A , 065201 (2020).[8] A. Okounkov, “Toda equations for Hurwitz numbers”, Math. Res. Lett. , 447-453(2000).[9] A. Okounkov and R. Pandharipande, “Gromov-Witten theory, Hurwitz theory andcompleted cycles,” Ann. Math. , 517-560 (2006)., 517-560 (2006).