Narrow Results

This paper aims at explaining some incarnations of the idea of topological recursion: in two-dimensional quantum field theories (2d TQFTs), in cohomological field theories (CohFT), and in the computation of volumes of the moduli space of curves. It gives an introduction to the formalism of quantum Airy structures on which the topological recursion is based, which is seen at work in the above topics.

We consider an elementary, and largely unexplored, combinatorial problem in low-dimensional topology: for a compact surface $S$, with a finite set of points $F$ fixed on its boundary, how many configurations of disjoint arcs are there on $S$ whose boundary is $F$? We find that this enumerative problem, counting curves on surfaces, has a rich structure. We show that such curve counts obey an effective recursion, in the general spirit of topological recursion, and exhibit quasi-polynomial behavior. This “elementary curve-counting” is in fact related to a more advanced notion of “curve-counting” from algebraic geometry or symplectic geometry. The asymptotics of this enumerative problem are closely related to the asymptotics of volumes of moduli spaces of curves, and the quasi-polynomials governing the enumerative problem encode intersection numbers on moduli spaces. Among several other results, we show that generating functions and differential forms for these curve counts exhibit structure that is reminiscent of the mathematical physics of free energies, partition functions and quantum curves.

We prove a Givental type decomposition for partition functions that arise out of topological recursion applied to spectral curves. Copies of the Konstevich–Witten KdV tau function arise out of regular spectral curves and copies of the Brezin–Gross–Witten KdV tau function arise out of irregular spectral curves. We present the example of this decomposition for the matrix model with two hard edges and spectral curve $(x^2-4)y^2=1$.

The purpose of this paper is to study the eigenvalues of U^t where U is a large N×N random unitary matrix and t > 0. In particular we are interested in the typical times t for which all the eigenvalues are simultaneously close to 1 in different ways thus corresponding to recurrence times in the issue of quantum measurements. Our strategy consists in rewriting the problem as a random matrix integral and use loop equations techniques to compute the first-orders of the large N asymptotic. We also connect the problem to the computation of a large Toeplitz determinant whose symbol is the characteristic function of several arc segments of the unit circle. In particular in the case of a single arc segment we recover Widom's formula. Eventually we explain why the first return time is expected to converge toward an exponential distribution when N is large. Numerical simulations are provided along the paper to illustrate the results.

This chapter aims at giving an introduction to the notion of quantum curves. The main purpose is to describe the new discovery of the relation between the following two disparate subjects: one is the topological recursion, that has its origin in random matrix theory and has been effectively applied to many enumerative geometry problems; and the other is the quantization of Hitchin spectral curves associated with Higgs bundles. Our emphasis is on explaining the motivation and examples. Concrete examples of the direct relation between Hitchin spectral curves and enumeration problems are given. A general geometric framework of quantum curves is also discussed.