Witten conjecture

Suppose that M_g,n is the moduli stack of compact Riemann surfaces of genus g with n distinct marked points x₁,...,x_n, and M_g,n is its Deligne–Mumford compactification. There are n line bundles L_i on 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 〈τ_d1, ..., τ_dn〉 is the intersection index of Π c₁(L_i)^d_i on M_g,n where Σd_i = dimM_g,n = 3g – 3 + n, and 0 if no such g exists, where c₁ is the first Chern class of a line bundle. Witten's generating function

${\displaystyle F(t_{0},t_{1},\ldots )=\sum \langle \tau _{0}^{k_{0}}\tau _{1}^{k_{1}}\cdots \rangle \prod _{i\geq 0}{\frac {t_{i}^{k_{i}}}{k_{i}!}}={\frac {t_{0}^{3}}{6}}+{\frac {t_{1}}{24}}+{\frac {t_{0}t_{2}}{24}}+{\frac {t_{1}^{2}}{24}}+{\frac {t_{0}^{2}t_{3}}{48}}+\cdots }$

encodes all the intersection indices as its coefficients.

Witten's conjecture states that the partition function Z = exp F is a τ-function for the KdV hierarchy, in other words it satisfies a certain series of partial differential equations corresponding to the basis ${\displaystyle \{L_{-1},L_{0},L_{1},\ldots \}}$ of the Virasoro algebra.

Kontsevich used a combinatorial description of the moduli spaces in terms of ribbon graphs to show that

${\displaystyle \sum _{d_{1}+\cdots +d_{n}=3g-3+n}\langle \tau _{d_{1}},\ldots ,\tau _{d_{n}}\rangle \prod _{1\leq i\leq n}{\frac {(2d_{i}-1)!!}{\lambda _{i}^{2d_{i}+1}}}=\sum _{\Gamma \in G_{g,n}}{\frac {2^{-|X_{0}|}}{|{\text{Aut}}\Gamma |}}\prod _{e\in X_{1}}{\frac {2}{\lambda (e)}}}$

Here the sum on the right is over the set G_g,n of ribbon graphs X of compact Riemann surfaces of genus g with n marked points. The set of edges e and points of X are denoted by X ₀ and X₁. The function λ is thought of as a function from the marked points to the reals, and extended to edges of the ribbon graph by setting λ of an edge equal to the sum of λ at the two marked points corresponding to each side of the edge.

By Feynman diagram techniques, this implies that F(t₀,...) is an asymptotic expansion of

${\displaystyle \log \int \exp(i{\text{tr}}X^{3}/6)d\mu }$

as Λ lends to infinity, where Λ and Χ are positive definite N by N hermitian matrices, and t_i is given by

${\displaystyle t_{i}={\frac {-{\text{tr }}\Lambda ^{-1-2i}}{1\times 3\times 5\times \cdots \times (2i-1)}}}$

and the probability measure μ on the positive definite hermitian matrices is given by

${\displaystyle d\mu =c_{\Lambda }\exp(-{\text{tr}}X^{2}\Lambda /2)dX}$

where c_Λ is a normalizing constant. This measure has the property that

${\displaystyle \int X_{ij}X_{kl}d\mu =\delta _{il}\delta _{jk}{\frac {2}{\Lambda _{i}+\Lambda _{j}}}}$

which implies that its expansion in terms of Feynman diagrams is the expression for F in terms of ribbon graphs.

From this he deduced that exp F is a τ-function for the KdV hierarchy, thus proving Witten's conjecture.

The Witten conjecture is a special case of a more general relation between integrable systems of Hamiltonian PDEs and the geometry of certain families of 2D topological field theories (axiomatized in the form of the so-called cohomological field theories by Kontsevich and Manin), which was explored and studied systematically by B. Dubrovin and Y. Zhang, A. Givental, C. Teleman and others.

The Virasoro conjecture is a generalization of the Witten conjecture.

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