aa r X i v : . [ m a t h . AG ] D ec AN EXPLICIT FORMULA FOR WITTEN’S 2-CORRELATORS
PETER ZOGRAF
Abstract.
An explicit closed form expression for 2-correlators of Witten’stwo dimensional topological gravity is derived in arbitrary genus.
Let M g,n be the Deligne-Mumford moduli space of genus g complex stablealgebraic curves with n > L i → M g,n , i = 1 , . . . , n . Recall that L i is defined fiberwise by L i | C,x ,...,x n ∼ = T ∗ x i C , where C is a genus g curve with marked points x , . . . , x n .Put ψ i = c ( L i ) , i = 1 , . . . , n, and, following Witten [5], define h τ d . . . τ d n i = Z M g,n ψ d . . . ψ d n n , where d + . . . + d n = 3 g − n (we assume that h τ d . . . τ d n i = 0 if any of d i < h τ d . . . τ d n i are called correlators of Witten’s twodimensional topological gravity. The famous Witten’s conjecture [5] (Kontsevich’stheorem [3]) claims that the generating function of these numbers (free energy of twodimensional topological gravity) satisfies the KdV (Korteveg-deVries) hierarchy.Computability of the intersection numbers h τ d . . . τ d n i is an important problem.For g = 0 by a result of Kontsevich [3] h τ d . . . τ d n i = ( n − d ! . . . d n !(i. e. a multinomial coefficient). For g = 1, closed form expressions for the inter-section numbers as sums of the multinomial coefficients were obtained, e. g., in [2].However, no general explicit formulas for the numbers h τ d . . . τ d n i are known for g > h τ k τ g − − k i , for arbitrary g (in this case n = 2 and k ranges from0 to 3 g − g − n ) h τ k n Y i =1 τ d i i = (2 k + 3) h τ τ k +1 n Y i =1 τ d i i− h τ τ k n Y i =1 τ d i i − X I ∪ J = { ,...,n } h τ τ k Y i ∈ I τ d i ih τ Y j ∈ J τ d j i . Adapted for n = 1 (the case of 2-correlators), it reads2 g h τ k τ g − − k i =(2 k + 3) h τ τ k +1 τ g − − k i − h τ τ k τ g − − k i − h τ τ k ih τ τ g − − k i . (1)We will use the string equation h τ n Y i =1 τ d i i = n X j =1 h n Y i =1 τ d i − δ ij i (2) This work was supported by the Russian Science Foundation grant 16-11-10039. and the dilaton equation h τ n Y i =1 τ d i i = (2 g − n ) h n Y i =1 τ d i i (3)for Witten’s correlators, see [5]. Applying the string equation (2) to (1), we easilyget that(2 k + 3) h τ k +1 τ g − − k i = (2 g − − k ) h τ k τ g − − k i + 16 ( h τ k − τ g − − k i + 3 h τ k − τ g − − k i + 3 h τ k − τ g − − k i + h τ k τ g − − k i )+ h τ k − ih τ g − − k i . (4)From here, using (2), (3) and the fact from [5] that h τ g − i = 124 g g ! , one can recursively compute 2-correlators h τ k τ g − − k i for any g ≥ k =0 , . . . , g − a g,k = (2 k + 1)!!(6 g − − k )!! h τ k τ g − − k i (6 g − h τ τ g − i = (2 k + 1)!!(6 g − − k )!!(6 g − g g ! h τ k τ g − − k i . (5)Then in terms of a g,k we can rewrite (4) as follows:(6 g − − k ) a g,k +1 = (2 g − − k ) a g,k + 4 g (6 g − g − g −
5) ((2 k + 1)(2 k − k − a g − ,k − + 3(2 k + 1)(2 k − g − − k ) a g − ,k − + 3(2 k + 1)(6 g − − k )(6 g − − k ) a g − ,k − +(6 g − − k )(6 g − − k )(6 g − − k ) a g − ,k )+ g ! j !( g − j )! (2 k +1)!!(6 g − − k )!!(6 g − , k = 3 j − , , otherwise . (6)Using (2) and (3), it is elementary to show that a g, = 1 , a g, = 6 g − g − . Consider now the differences b g,k = a g,k +1 − a g,k , k = 0 , . . . , (cid:2) g − (cid:3) −
1. Below wederive simple explicit formulas for these numbers. Actually, we have the following
Lemma.
The numbers b g,k are explicitly given by the formulas b g,k = (6 g − − k )!!(6 g − · (6 j − j ! ( g − g − j )! ( g − j ) , k = 3 j − , − (6 j +1)!! j ! ( g − g − − j )! , k = 3 j, (6 j +3)!! j ! ( g − g − − j )! , k = 3 j + 1 . (7) Proof.
Take the difference(6 g − − k ) a g,k +2 − (6 g − − k ) a g,k +1N EXPLICIT FORMULA FOR WITTEN’S 2-CORRELATORS 3 and apply formula (6) to both of its terms. A straightforward computation yieldsthe following recursion for b g,k +1 = a g,k +2 − a g,k +1 :(6 g − − k ) b g,k +1 = (2 g − − k ) b g,k + 4 g (6 g − g − g −
5) ((2 k + 1)(2 k − k − b g − ,k − + 3(2 k + 1)(2 k − g − − k ) b g − ,k − + 3(2 k + 1)(6 g − − k )(6 g − − k ) b g − ,k − +(6 g − − k )(6 g − − k )(6 g − − k ) b g − ,k )+ g ! j !( g − j )! (2 k +3)!!(6 g − − k )!!(6 g − , k = 3 j − , − g ! j !( g − j )! (2 k +1)!!(6 g − − k )!!(6 g − , k = 3 j − , , k = 3 j. (8)Now take the values of b g,k given by (7) and substitute them into (8). After a lengthybut elementary computation we see that the numbers b g,k satisfy the recursion (8).This completes the proof since the recursion (8) has a unique solution with giveninitial values. (cid:3) By the definition of the numbers b g,k we have a g,k = a g, + k − X i =1 b g,i , k = 2 , . . . , (cid:20) g − (cid:21) − . Together with (5) this yields
Theorem.
The following closed form expression is valid for the 2-correlators: h τ k τ g − − k i = (6 g − g g !(2 k + 1)!!(6 g − − k )!! g − g − k − X i =1 b g,i ! . Here the numbers b g,i are given by formula (7) , and k = 2 , . . . , (cid:2) g − (cid:3) − .Remark. A careful analysis of the numbers b g,k performed in [1], Sect. 4, impliesthat for any g ≥ k = 2 , . . . , g − g − g − < a g,k < . Acknowledgements.
The author thanks A. Zorich for stimulating discussions.
References [1] V. Delecroix, E. Goujard, P. Zograf, A. Zorich, Masur-Veech volumes, frequencies of simpleclosed geodesics and intersection numbers of moduli spaces of curves, arXiv:2011.05306 (2020)(to appear in Duke Math. J.)[2] A. Kabanov, T. Kimura, Intersection numbers and rank one cohomological field theories ingenus one, Commun. Math. Phys. (1998), 651-674.[3] M. Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy func-tion, Commun. Math. Phys. (1992), 1-23.[4] K. Liu, H. Xu, An effective recursion formula for computing intersection numbers,arXiv:0710.5322 (2007).[5] E. Witten, Two-dimensional gravity and intersection theory on moduli space, Surveys in Diff.Geom. (1991), 243-310. St.Petersburg Department of the Steklov Mathematical Institute, Fontanka 27,St. Petersburg 191023 Russia, and Chebyshev Laboratory of St. Petersburg StateUniversity, 14th Line V.O. 29B, St.Petersburg 199178 Russia
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