Source: JMM 2025

Doodling

I want to remember that if we doodle around a diagram it eventually becomes more and more like a circle.

Volume of $n$-sphere

Witten's Conjecture

Suppose that $M_{g,n}$ is the moduli stack of compact Riemann surfaces of genus $g$ with $n$ distinct marked points $x_1, \dots, x_n$, and $\overline{M}_{g,n}$ is its Deligne–Mumford compactification. There are $n$ line bundles $L_i$ on $\overline{M}_{g,n}$, whose fiber at a point of the moduli stack is given by the cotangent space of a Riemann surface at the marked point $x_i$. The intersection index $\langle \tau_{d_1}, \dots, \tau_{d_n} \rangle$ is the intersection index of $$ \prod_{i=1}^n c_1(L_i)^{d_i} $$ on $\overline{M}_{g,n}$, where $\sum_{i=1}^n d_i = \dim \overline{M}_{g,n} = 3g - 3 + n$, and 0 if no such $g$ exists, where $c_1$ is the first Chern class of a line bundle.

Witten's generating function: $$ F(t_0,t_1,\dots) = \sum \langle \tau_0^{k_0} \tau_1^{k_1} \cdots \rangle \prod_{i \geq 0} \frac{t_i^{k_i}}{k_i!} = \frac{t_0^3}{6} + \frac{t_1}{24} + \frac{t_0 t_2}{24} + \frac{t_1^2}{24} + \frac{t_0^2 t_3}{48} + \cdots $$ encodes all the intersection indices as its coefficients.

Witten's Conjecture

The partition function $Z = \exp F$ is a $\tau$-function for the KdV hierarchy, in other words, it satisfies a certain series of partial differential equations corresponding to the basis ${ L_{-1}, L_0, L_1, \dots }$ of the Virasoro algebra.