A Remark on Equivalence between Two Formulas of the Two point Witten-Kontsevich Correlators
aa r X i v : . [ m a t h - ph ] F e b A Remark on Equivalence between Two Formulas ofthe Two-point Witten-Kontsevich Correlators
Jindong Guo
Abstract
We prove the equivalence between two explicit expressions for two-point Witten-Kontsevich correlators obtained by M. Bertola, B. Dubrovin, D. Yang [1] and byP. Zograf [8].
Let M g,n denote the Deligne–Mumford moduli space of stable curves of genus g with n marked points. The genus g intersection numbers of ψ -classes h τ d τ d . . . τ d n i g are defined by h τ d τ d . . . τ d n i g = Z M g,n ψ d . . . ψ d n n Here ψ i denote the first Chern classes of the cotangent line bundles on M g,n . Thesenumbers, also called the n -point Witten-Kontsevich correlators due to their relationshipto the KdV hierarchy, vanish unless d + · · · + d n = 3 g + n −
3. For n = 2, M. Bertola,B. Dubrovin and D. Yang [1] (see page 45 therein) derives an explicit expression of thetwo-point Witten-Kontsevich correlators h τ d τ d i g = g P l =0 (3 g +2 − l ) a l − , g − l (6 g +3)!!(6 g +3)!! d = 3 g + 1 , d = 3 g + 1 g P l =0 (3 g +3 − l ) a l − , g − l (6 g +5)!!(6 g +1)!! d = 3 g + 2 , d = 3 g (1)where d + d = 3 g −
1, and a k ,k is defined by a k ,k = (6 g − g − · g g − · ( g − g − k = 3 g − , k = 3 g − , g , g ≥ − (6 g − g − g g · g ! g ! 6 g +16 g − k = 3 g , k = 3 g − , g , g ≥ − (6 g − g − g g · g ! g ! 6 g +16 g − k = 3 g − , k = 3 g , g , g ≥ otherwise. (2)Another formula of the two-point Witten-Kontsevich correlators is obtained by P. Zo-graf [8], that reads as follows: h τ k τ g − − k i g = (6 g − g g !(2 k + 1)!!(6 g − − k )!! (cid:18) g − g − k − X i =1 b g,i (cid:19) (3)1here b g,k is defined by b g,k = (6 g − − k )!!(6 g − · (6 j − g − j !( g − j )! ( g − j ) k = 3 j − − (6 j +1)!!( g − j !( g − − j )! k = 3 j (6 j +3)!!( g − j !( g − − j )! k = 3 j + 1 (4)Our goal is to prove directly that the two formulas (1) and (3) are equivalent. Ourproof is based on an identity between the numbers a k ,k and b g,k , given by the followinglemma. Lemma.
The following identity is true: (6 g − g g ! b g,k = k +1 X l =0 a l − , g − l (5) Proof.
For k = −
1, the identity is trivial. By direct computation one finds(6 g − g g ! ( b g,k − b g,k − ) = a k +1 , g − k − (6)where the computation has three cases according to the value of k (mod 3), and byinduction the lemma follows.Let k =3 g + 1, g = g + g + 1 and k =3 g + 2, g = g + g + 1, we have Theorem.
The two formulas (1) and (3) are equivalent.Proof.
The theorem is equal to the following explicit equalities: g +1 P l =0 (3 g + 2 − l ) a l − , g − l (6 g + 3)!!(6 g − g − g − g g !(6 g + 3)!!(6 g − g − (cid:18) g − g − g X i =1 b g,i (cid:19) (7) g +2 P l =0 (3 g + 3 − l ) a l − , g − l (6 g + 5)!!(6 g − g − g − g g !(6 g + 5)!!(6 g − g − (cid:18) g − g − g +1 X i =1 b g,i (cid:19) (8)We begin with proving the first equality (7). By a straightforward calculation, we findthat it suffices to prove g +1 X l =0 (3 g + 2 − l ) a l − , g − l = (6 g − g g ! g X i = − b g,i (9)Observing the following elementary equalities: g +1 X l =0 (3 g + 2 − l ) a l − , g − l = g +1 X l =0 l X i =0 a i − , g − i = g X l = − l +1 X i =0 a i − , g − i and using the Lemma, we find that equality (9) is true. The proof for the secondequality (8) is similar. 2n a subsequent publication, we prove in a direct way the equivalence between the n -point functions considered in this paper [1, 3, 6, 7, 8] and the n -point functionsconsidered by Buryak, Dijkgraaf, Liu–Xu, Okounkov, Zagier (cf. [2, 4, 5]); the explicitrelationship between these two types of n -point functions can be found in Section 8of [1]. Acknowledgements.
The author thanks Di Yang for his advice.
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