Moduli spaces of connections and spectral curves: WKB or Topological Recursion, the role of integrable systems...

par Bertrand Eynard (CEA Saclay)

jeudi 10 sept. 2026, 14:00 → 16:00 Europe/Paris

Understanding the geometry of moduli spaces of meromorphic (with poles) connections on $G$-principal bundles over a Riemann surface, is a deep challenging problem in mathematics with a large range of applications. Let $(P,\nabla)$ a G-bundle with a connection $\nabla = d-A(x)$, and a flat section $\Psi(x)$ satisfying $\nabla\Psi=0$, i.e. a differential system.

A useful way to study it, is to rescale $A\to 1/\epsilon A$, i.e. the limit of "large connection field". One can study $\Psi(x,\epsilon)$ in that limit, through the WKB asymptotic expansion: $\Psi(x,\epsilon) \sim V(x) (1+O(\epsilon)) e^{1/\epsilon \int^x Y(x')} $ where $VYV^{-1}=A$ is the diagonalization of $A$, i.e. in a matrix representation, $Y(x) = \operatorname{diag}(Y_1(x),\dots,Y_r(x))$, is the eigenvaleus matrix, each $Y_i(x)$ is on the spectral curve $P(x,y)=\det(y-A(x))=0$.

In other words, to each connection, one can associate a spectral curve $P(x,y)=0$, and from there, solving WKB recursively, reconstruct the full $\Psi$ as a formal series in $\epsilon$.

This gives a map between the moduli-space of connections, and the moduli space of spectral curves (Hitchin's base). This map is well studied in the non-Abelian Hodge correspondence.

However, in mathematical physics, in integrable systems, there is another map between the same spaces, and which has more beautiful properties, in particular it satisfies an integrable system, it satisfies Hirota equations, Seiberg-Witten equations, Miwa-Jimbo equations, modular properties and many other beautiful properties.

This map is the one relevant for almost all interesting mathematical physics problems, like random matrices, enumerative geometry, combinatorics of maps, Gromov-Witten theory, Jones polynomials...

The inverse map doesn't use WKB, instead, starting from a spectral curve, it computes $\Psi(x,\epsilon)$ by a universal recursion in the coefficients of powers of $\epsilon$: the Topological Recursion.

Therefore, Topological Recursion provides a map: moduli-space of Spectral curves -> moduli space of connections.

 

In this introductory pedagogical talk, we shall introduce these notions by simple examples, like random matrices, and see how the map is constructed and how it obeys beautiful properties.