Witten conjecture

In algebraic geometry, the Witten conjecture is a conjecture about intersection numbers of stable classes on the moduli space of curves, introduced by Edward Witten in the paper Witten (1991), and generalized in Witten (1993). Witten's original conjecture was proved by Maxim Kontsevich in the paper Kontsevich (1992).

Witten's motivation for the conjecture was that two different models of 2-dimensional quantum gravity should have the same partition function. The partition function for one of these models can be described in terms of intersection numbers on the moduli stack of algebraic curves, and the partition function for the other is the logarithm of the τ-function of the KdV hierarchy. Identifying these partition functions gives Witten's conjecture that a certain generating function formed from intersection numbers should satisfy the differential equations of the KdV hierarchy.

Statement

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.

Proof

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.

Generalizations

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.

References

• Cornalba, Maurizio; Arbarello, Enrico; Griffiths, Phillip A. (2011), Geometry of algebraic curves. Volume II, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 268, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-540-69392-5, ISBN 978-3-540-42688-2, MR 2807457
• Kazarian, M. E.; Lando, Sergei K. (2007), "An algebro-geometric proof of Witten's conjecture", Journal of the American Mathematical Society, 20 (4): 1079–1089, arXiv:math/0601760, Bibcode:2007JAMS...20.1079K, doi:10.1090/S0894-0347-07-00566-8, ISSN 0894-0347, MR 2328716
• Kontsevich, Maxim (1992), "Intersection theory on the moduli space of curves and the matrix Airy function", Communications in Mathematical Physics, 147 (1): 1–23, Bibcode:1992CMaPh.147....1K, doi:10.1007/BF02099526, ISSN 0010-3616, MR 1171758
• Lando, Sergei K.; Zvonkin, Alexander K. (2004), Graphs on surfaces and their applications (PDF), Encyclopaedia of Mathematical Sciences, 141, Berlin, New York: Springer-Verlag, ISBN 978-3-540-00203-1, MR 2036721
• Witten, Edward (1991), "Two-dimensional gravity and intersection theory on moduli space", Surveys in differential geometry (Cambridge, MA, 1990), 1, Bethlehem, PA: Lehigh Univ., pp. 243–310, ISBN 978-0-8218-0168-0, MR 1144529
• Witten, Edward (1993), "Algebraic geometry associated with matrix models of two-dimensional gravity", in Goldberg, Lisa R.; Phillips, Anthony V. (eds.), Topological methods in modern mathematics (Stony Brook, NY, 1991), Proceedings of the symposium in honor of John Milnor's sixtieth birthday held at the State University of New York, Stony Brook, New York, June 14–21, 1991., Houston, TX: Publish or Perish, pp. 235–269, ISBN 978-0-914098-26-3, MR 1215968