Formal pseudodifferential operators and Witten’s r-spin numbers

Abstract

We derive an effective recursion for Witten's r-spin intersection numbers, using Witten's conjecture relating r-spin numbers to the Gel'fand-Dikii hierarchy (Theorem 4.1). Consequences include closed-form descriptions of the intersection numbers (for example, in terms of gamma functions: Propositions 5.2 and 5.4, Corollary 5.5). We use these closed-form descriptions to prove Harer-Zagier's formula for the Euler characteristic of M g,1 . Finally in §6, we extend Witten's series expansion formula for the Landau-Ginzburg potential to study r-spin numbers in the small phase space in genus zero. Our key tool is the calculus of formal pseudodifferential operators, and is partially motivated by work of Brézin and Hikami.

From r-spin intersection numbers to Hodge integrals

“…The listed results in (6.14) are coincided with the ones which have also been computed by K.Liu and his collaborators [33]. However, if we want to derive certain intersection numbers listed in [33], we need to compute more schur polynomials. The calculation is beyond the capability of our computers.…”

“…The calculation is beyond the capability of our computers. On the other side, in our approach, while it is easy to compute certain intersection numbers which is not listed in [33]. τ 2 0,0 τ 0,3 0 = 1, τ 0,0 τ 0,1 τ 0,2 0 = 1, τ 3 0,1 0 = 1, τ 3 0,0 τ 1,3 0 = 1 6 , τ 2 0,0 τ 0,3 τ 1,0 0 = 7 6 , τ 0,0 τ 2 0,1 τ 1,1 0 = 2, τ 0,0 τ 0,1 τ 0,2 τ 1,0 0 = 7 6 , τ 3 0,1 τ 1,0 0 = 7 6 , τ 2 0,1 τ 2 0,3 0 = 1 20 , τ 0,1 τ 2 0,2 τ 0,3 0 = 1 5 , τ 4 0,2 0 = 2 5 , τ 0,0 τ 2,0 1 = 1 6 , τ 0,0 τ 1,3 τ 1,2 1 = 1 30 , τ 2 1,0 1 = 7 36 , τ 2 0,0 τ 3,0 1 = 1 12 , τ 0,0 τ 0,2 τ 2,3 1 = 1 60 , τ 0,0 τ 0,3 τ 2,2 1 = 1 60 , τ 0,0 τ 1,0 τ 2,0 1 = 1 3 , τ 0,1 τ 1,1 τ 1,3 1 = 1 20 , τ 0,3 τ 2 1,1 1 = 1 40 , τ 3 1,0 1 = 1 18 , τ 1,0 τ 3,2 2 = 11 1200 , · · · · · · A.5 r = 7…”

“…After taking the logarithm of Z {r} ∞ [M] and resplacing the parameter λ by t m,a → λ r(m−1)(r+1) + a r+1 t m,a , we can get the genus expansion of the free energy which is the generating function of the r-spin intersection numbers. Unlike the recursive method given by K.Liu, R.Vakil and H.Xu[33], we completely solve a generating function of r-spin intersection numbers. From this generating function, we can read all the r-spin intersection numbers directly.…”

Punctures and p-spin curves from matrix models III. Dl type and logarithmic potential

“…This reformulation enables us to obtain easily the intersection numbers for integer p (Neveu-Schwarz punctures) for n point functions, which should be consistent with the results obtained by the recursive method [10,11,13] due to Gelfand-Dikii equation. The evaluation of several marked points in general p and for genus g was obtained in the recursive calculations [11], and we show in this article that our method of the Laurent expansion agrees with them for the lower orders, especially for three point function.…”

Simple Lie Algebras and Topological ODEs

“…Then, from (5.12) we expect that all one-point r-spin intersection numbers can be read off from coefficients of solutions to the n th dominant ODE expanded near the regular singularity x = 0. We also remark that alternative closed expressions for one-point r-spin intersection numbers have been obtained in [28] by using the Gelfand-Dickey pseudo-differential operators.…”