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New results of intersection numbers on moduli spaces of curves

Edited by ST. Yau, Harvard University, Cambridge, MA, and approved July 13, 2007 (received for review June 23, 2007)
Abstract
We present a series of results we obtained recently about the intersection numbers of tautological classes on moduli spaces of curves, including a simple formula of the npoint functions for Witten's τ classes, an effective recursion formula to compute higher Weil–Petersson volumes, several new recursion formulae of intersection numbers and our proof of a conjecture of Itzykson and Zuber concerning denominators of intersection numbers. We also present Virasoro and KdV properties of generating functions of general mixed κ and ψ intersections.
Let denote the Deligne–Mumford moduli stack of stable curves of genus g with n marked points. Let ψ_{i} be the first Chern class of the line bundle 𝔼, whose fiber over each pointed stable curve is the cotangent line at the ith marked point. Let λ_{i} be the ith Chern class of the Hodge bundle 𝔼, whose fiber over each pointed stable curves is H ^{0} (C, ω_{c}).
We also have the κ classes originally defined by Mumford (1), Morita (2), and Miller (3). A more natural variation was later given by Arbarello–Cornalba (4). It is known that the κ and ψ classes generate the tautological cohomology ring of the moduli spaces, and most of the known cohomology classes are tautological.
The following intersection numbers are called the higher Weil–Petersson volumes (5). These are important invariants of moduli spaces of curves.
In 1990, Witten (6) made the remarkable conjecture that the generating function of intersection numbers of ψ classes on moduli spaces are governed by KdV hierarchy. Witten's conjecture (first proved by Kontsevich; ref. 7) is among the deepest known properties of moduli spaces of curves and motivated a surge of subsequent developments.
The intersection theory of tautological classes on the moduli space of curves is a very important subject and has close connections to string theory, quantum gravity and many branches of mathematics.
The nPoint Functions for Intersection Numbers
Definition 1:
We call the following generating function the npoint function.
Consider the following “normalized” npoint function Starting from 1point function G(x) = 1/x ^{2}, we can obtain any npoint function recursively by the following theorem.
Theorem 1 (8).
For n ≥ 2, where P _{r} and Δ are homogeneous symmetric polynomials defined by where I, J ≠ θ, ṉ = {1, 2, …, n}, and G_{g} (x_{I}) denotes the degree 3g + I − 3 homogeneous component of the normalized Ipoint function G(x_{k1} , …, x _{kI}), where k_{j} ∈ I.
Thus, we have an elementary and more efficient algorithm to calculate all intersection numbers of ψ classes other than the celebrated Witten–Kontsevich theorem.
Because for r > 0 and we recover Dijkgraaf's 2point function and Zagier's 3point function obtained years ago.
There is another slightly different formula of the npoint functions. When n = 3, this has also been obtained by Zagier.
Theorem 2 (8).
For n ≥ 2, where P _{r} and Δ are the same polynomials as defined in Theorem 2.
Okounkov (9) obtained an analytic expression of the npoint functions using ndimensional errorfunctiontype integrals. Brézin and Hikami (10) use correlation functions of GUE ensemble to find explicit formulae of npoint functions.
Higher Weil–Petersson Volumes
We have discovered a general recursion formula of higher Weil– Petersson volumes (11), which is a vast generalization of the Mirzakhani's recursion formula (12).
First we fix notations as in ref. 5. Consider the semigroup N ^{∞} of sequences m = (m _{1}, m _{2}, …), where m _{i} are nonnegative integers and m _{i} = 0 for sufficiently large i.
Let m, t, a_{1}, …, a_{n} ∈ N ^{∞} and s := (s _{1}, s _{2}, …) be a family of independent formal variables. Let b ∈ N ^{∞}, we denote a formal monomial of κ classes by
Theorem 3 (11).
Let b ∈ N ^{∞} and d _{j} ≥ 0. where the tautological constants α_{L} can be determined recursively from the following formula namely with the initial value α_{0} = 1.
The proof of the above theorem is to use Witten–Kontsevich theorem, a combinatorial formula in ref. 5 expressing κ classes by ψ classes and the following elementary but crucial lemma (11).
Lemma 1.
Let F (L, n) and G (L, n) be two functions defined on N^{∞} × ℕ, where ℕ = {0, 1, 2,…} is the set of nonnegative integers. Let α_{L} and β_{L} be real numbers depending only on L ∈ N^{∞} that satisfy α_{0}β_{0} = 1 and Then the following two identities are equivalent. When b = (l, 0, 0, …), Theorem 3 recovers Mirzakhani's recursion formula of Weil–Petersson volumes for moduli spaces of bordered Riemann surfaces (12–17).
Theorem 3 also provides an effective algorithm to compute higher Weil–Petersson volumes recursively.
In fact, we can use the main formula in ref. 5 to generalize almost all pure ψ intersections to identities of higher Weil–Petersson volumes that share similar structures as Theorem 3. For example, the identities in the following theorem are generalizations of the string and dilation equations.
Theorem 4 (11).
For b ∈ N ^{∞} and d _{j} ≥ 0, and Note that Theorem 4 generalizes the results in ref. 18.
New Identities of Intersection Numbers
The next two theorems follow from a detailed study of coefficients of the npoint functions in Theorem 1.
Theorem 5 (8).
We have

Let k > 2g, d_{j} ≥ 0 and Σ_{j=1} ^{n} d_{j} = 3g + n − k.

Let d_{j} ≥ 1 and Σ_{j=1} ^{n} d_{j} = g + n.
Theorem 6 (8, 19).
We have

Let k > 2g, d_{j} ≥ 0 and Σ_{j=1} ^{n} d_{j} = 3g + n − k − 1.

Let d_{j} ≥ 1 and Σ_{j=1} ^{n} (d_{j} −1) = g − 1.
In fact, it's easy to see that Theorems 5 and 6 imply each other through the following proposition.
Proposition 7 (8).
Let d_{j} ≥ 0 and Σ_{j=1} ^{n} d_{j} = g + n.
Because ch_{k} (𝔼) = 0 for k > 2g, λ_{g}λ_{g−1} = (−1)^{g−1} (2g − 1)!ch_{2g−1} (𝔼), by Mumford's formula (1) of the Chern character of Hodge bundles, it's not difficult to see that Theorem 6 implies the following theorem.
Theorem 8 (8, 19).
Let k be an even number and k ≥ 2g, d_{j} ≥ 0, Σ_{j=1} ^{n} d_{j} = 3g + n − k − 2. Note that, when k = 2g, the above theorem is equivalent to the following Hodge integral identity (20) (also known as Faber's intersection number conjecture; ref. 21) where Σ_{j=1} ^{n} (d_{j} −1) = g − 2 and d_{j} ≥ 1.
The above λ_{g}λ_{g−1} integral follows from degree 0 Virasoro constraints for ℙ^{2} announced by Givental (22). However it is very desirable to have a direct proof of Theorem 8 when k = 2g, possibly using our explicit formulae of the npoint functions (see also ref. 23).
As pointed out in the last section, we can generalize all of the above new recursion formulae of ψ classes to identities of higher Weil–Petersson volumes. For example, we may generalize Proposition 7 and Theorem 8 to the following.
Proposition 9 (11).
Let b ∈ N ^{∞}, d_{j} ≥ 0.
Theorem 10 (11).
Let b ∈ N ^{∞}, M ≥ 2g be an even number and d_{j} ≥ 0.
We also found the following conjectural identity experimentally, which is amazing if compared with Theorems 6 and 8.
Conjecture 11 (19).
Let g ≥ 2, d_{j} ≥ 1, Σ_{j=1} ^{n} (d_{j} −1) = g.
Because (2g − 3)!ch_{2g−3}(𝔼) = (−1)^{g−1} (3λ_{g−3}λ_{g} − λ_{g−1}λ_{g−2}), it's easy to see that the above identity is equivalent to the following identity of Hodge integrals.
Conjecture 12.
Let g ≥ 2, d_{j} ≥ 1, Σ_{j=1} ^{n} (d_{j} − 1) = g.
Virasoro Constraints and KdV Hierarchy
From Theorem 3, we found new Virasoro constraints and KdV hierarchy for generating functions of higher Weil–Petersson volumes that vastly generalize the Witten conjecture and the results of Mulase and Safnuk (15).
Let s := (s _{1}, s _{2}, …) and t := (t _{0}, t _{1}, t _{2}, …), we introduce the following generating function where s^{m} = ∏i≥1 s _{i} ^{mi}.
We introduce the following family of differential operators for k ≥ −1, where γ_{L} are defined by
Theorem 13 (11, 15).
We have V_{k} exp(G) = 0 for k ≥ − 1 and the operators V_{k} satisfy the Virasoro relations
The Witten–Kontsevich theorem states that the generating function for ψ class intersections is a τfunction for the KdV hierarchy.
Because Virasoro constraints uniquely determine the generating functions G(s, t _{0}, t _{1}, …) and F(t _{0}, t _{1}, …), we have the following theorem.
Theorem 14 (11, 15).
where p_{k} are polynomials in s given by In particular, for any fixed values of s, G(s, t) is a τfunction for the KdV hierarchy.
Theorem 14 also generalized results in ref. 24.
Denominators of Intersection Numbers
Let denom(r) denote the denominator of a rational number r in reduced from (coprime numerator and denominator, positive denominator). We define and for g ≥ 2, where lcm denotes least common multiple.
Because denominators of intersection numbers on M̄_{g,n} all come from orbifold quotient singularities, the divisibility properties of D _{g,n} and _{g} should reflect overall behavior of singularities.
We have the following properties of D _{g,n} and _{g}.
Proposition 15 (25).
We have D_{g,n}  D _{g,n+1}, D_{g,n}  _{g} and _{g} = D_{g,3g−3} .
Theorem 16 (25).
For 1 < g′ ≤ g, the order of any automorphism group of a Riemann surface of genus g′ divides D _{g,3}.
The following corollary of Theorem 16 is a conjecture raised by Itzykson and Zuber (26) in 1992.
Corollary 17.
For 1 < g′ ≤ g, the order of any automorphism group of an algebraic curve of genus g′ divides _{g} .
The proof of Theorem 16 needs the following two lemmas (see ref. 25).
Lemma 2.
If p ≤ g + 1 is a prime number, then ord(p, D _{g,3}) ≥ 2.
Lemma 3 (27).
Let X be a Riemann Surface of genus g ≥ 2, then for any prime number p,
We have also obtained conjectural exact values of _{g} in ref. 19.
Acknowledgments
We thank ChiuChu Melissa Liu for helpful discussions.
Footnotes
 ^{‡}To whom correspondence should be addressed. Email: haoxu{at}cms.zju.edu.cn

Author contributions: K.L. and H.X. designed research, analyzed data, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.
 © 2007 by The National Academy of Sciences of the USA
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