Intersection numbers
The set of all Riemann surfaces (up to
conformal isomorphisms) of genus \(g\) with \(n\) marked points is called
the "moduli-space" \(M_{g,n}\). If \(2g-2+n>0\), \(M_{g,n}\) is an orbifold of
dimension \(3g-3+n\).
Moduli-spaces play a role in every string
theory, they constitute the simplest possible topological string
theory, as the space of worldsheets of given topology, with
target-space equal to a unique point. A goal is to understand the
topology of moduli spaces.
To the ith marked point \(p_i\), one
associates the 2-form \(\psi_i= c_1(T^*_{p_i})\) = the Chern class of the
cotangent plane at the point \(p_i\).
A product of \(3g-3+n\) Chern classes
is a top-dimensional form, and can be integrated over the full
moduli-space.
Introduced by Witten to compute amplitudes in
"topological gravity", integrals of Chern classes are called
"Intersection numbers", and are denoted (if \(d_1+...+d_n = 3g-3+n\) ):
\[\left<\tau_{d_1}\dots\tau_{d_n}\right>_g= \int_{\overline{M_{g,n}}} \psi_1^{d_1}\dots\psi_n^{d_n} \]
They are rational numbers, for example \(<\tau_1>_1=\int_{\overline{M_{1,1}}} \psi_1=\frac{1}{24} \) or \(<\tau_{14}\tau_2\tau_2>_6=\frac{379}{9555148800} \).
See
http://bertrand.eynard.free.fr/IN/IN.html
for computing intersection numbers.
The famous Witten's conjecture, proved by Kontsevich in 1991, was that the generating
function of intersection numbers, is the KdV Tau function, making
a link between Geometry and integrable systems.
Intersection numbers are topological invariants, they are the key to the
topology of the moduli spaces, they are the basic building blocks
of every string theory, and as Witten-Kontsevich have shown, they
are also the coefficients of expansions of KdV tau-functions and
play an ubiquitous role in every integrable systems, in fluid
mechanics, in random matrices...
They appear for instance as the
coefficients of the large \(x\) expansion of the log of the Airy
function
\[ \log(Ai(x)) = \frac{2}{3} x^{\frac32} -\frac{\log x}{4}
+ \sum_{n\geq 1} \sum_{g=0}^{\infty} \frac{1}{n!} (2x^{\frac32})^{2-2g-n}
\sum_{d_1+\dots+d_n=3g-3+n} < \tau_{d_1} \dots \tau_{d_n} >_g \prod_{i=1}^n ( 2d_i-1)!!
\]
\[ = \frac{2}{3} x^{\frac32} -\frac{\log x}{4}
+ \left( < \tau_{1} >_1 + \frac{1}{6} < \tau_{0}^3 >_0 \right)(2x^{\frac32})^{-1}
+ \left( \frac{1}{2} ( 6 < \tau_{0}\tau_2 >_1 + < \tau_{1}^2 >_1 ) + \frac4{24} < \tau_{0}^3 \tau_1 >_0 \right)(2x^{\frac32})^{-2} + O(x^{-\frac92})
\]
An issue however is how to compute them ?
As integrals
over \(M_{g,n}\) are undoable in practice, recursive methods based on
Virasoro algebra, or KdV equations were used.
We discovered an exact non-recursive formula, as sums over Young Tableaux:
\[
\left<\tau_{\lambda_1}\dots\tau_{\lambda_n}\right>_g
= \frac{2^{3g-3+n}}{24^g \prod_{i=1}^n (2\lambda_i+1)!!}
\sum_{|\mu|=3g-3+n} \left(
\sum_{r=0}^{\min(\frac12(n-1)(n-2),g)}
\sum_{|\nu|=3r-3+n} 12^r C_r(\nu)Q_{\nu,\mu} \right) \frac{K_{\mu,\lambda}}{\prod_{i=1}^n\prod_{j=1}^{\mu_i} (j-i+\frac32)^{-1}}
\]
where \(K_{\mu,\lambda}\) is the
Kostka number (number of
semi-standard Young tableaux of shape \(\lambda\) and weight \(\mu\) ), and
\(Q_{\nu,\mu} = \det \left(\frac{1}{( ((\mu_j-\nu_i+i-j)/3)!} \right)\)
(with the convention that factorial of non-integers is \(\infty\) ).
The coefficients \(C_r(\nu) \) are independent of the genus \(g\), they are given in our article
https://arxiv.org/abs/2212.04256.
For example with \(n=3\) we have \(C_0((0,0,0))=1\) and \(C_1((3,0,0))=\frac12\), i.e.
\[
\left<\tau_{\lambda_1}\tau_{\lambda_2}\tau_{\lambda_3}\right>_g
= \frac{2^{3g}}{24^g \prod_{i=1}^3 (2\lambda_i+1)!!}
\sum_{|\mu|=3g} (Q_{(0,0,0),\mu} + 6 Q_{(3,0,0),\mu}) K_{\mu,\lambda} \prod_{i=1}^3\prod_{j=1}^{\mu_i} (j-i+\frac32)
\]
Another novelty is the asymptotic formula for intersection numbers at large genus.
We proved the following formula
\[
< \tau_{d_1} \dots\tau_{d_n} >_g
\mathop{\sim}_{g\to\infty} \frac{ 2^n }{ 4\pi } \frac{ \Gamma(2g-2+n)}{(2/3)^{2g-2+n} \prod_{i} (2d_i+1)!! }
\left( 1+ \sum_{j=1}^k \frac{(2/3)^j}{(2g-3+n)^j} P_j +O(1/g^{k+1}) \right)
\]
and with an explicit computation of
all corrections \(P_j(n;n_0,n_1,n_2,\dots)\), which are polynomials in the numbers \(n_i\) of
degrees equal to \(i=0,1,2,\dots\), for example
\[ P_1(n;n_0) = -\frac14(n-n_0)^2 +\frac12(n-3)(n-n_0) - \frac{3n^2-15n+17}{12} \]
(the formula had been conjectured, only
to leading order).
Large genus assymptotics control the
non-perturbative effect in string theories, and in particular this
computation has brought high precision non-perturbative effects
for Jackiw-Teitelboim black-holes.
