Unique continuation through natural boundaries using resurgence

Resurgence theory is a wide-ranging extension of analyticity and of Borel summability, known to apply in many problems of natural origin. I will give an overview of some elements of this theory and discuss new results on unique continuation through natural boundaries, such as the unit circle for q-series, from a theoretical as well as numerical perspective. Work in collaboration with G. Dunne, A. Gruen and S. Gukov

Strong coupling results in N=2 gauge theories

We will discuss recent developments in the study of correlation functions of chiral scalar operators and integrated correlators in four-dimensional N=2 quiver gauge theories. Using supersymmetric localization, it is possible to map the computation of these observables to an interacting matrix model and obtain expressions that are valid for any value of the ’t Hooft coupling in the planar limit of the theory. In particular, we will focus on the strong-coupling regime, where these expressions allow us to compute the leading and subleading orders behavior of the observables in an analytic way

Modular resurgent structures of fermionic spectral traces on local P^2

In a joint project with C. Rella, we propose a new paradigm of modular resurgence that focuses on the role of the Stokes constants and the interplay of the their generating functions in form of q-series with the corresponding L-functions. Resurgent series with a modular resurgent structure are conjectured to have specific summability properties as well as to be closely related to quantum modular forms. Remarkably, a pivotal example arises from topological string theory, in the study of the resurgence structure of the first fermionic spectral traces of local P^2. In this talk, after discussing the general paradigm of modular resurgent series, I will focus on the example of local P^2. In particular, I will show that the generating functions of the Stokes constants (both in the weak and strong coupling regimes) are quantum modular forms. This talk is based on arxiv:2404.11550 and arxiv:2404.10695.

3d modularity revisited - defect 3-manifold invariants and VOAs

In this talk I will revisit some results regarding the modularity of a particular type of topological 3-manifold invariants, which were originally defined as q-series to categorify WRT invariants. These invariants are related to characters of logarithmic vertex operator algebras, and give examples of quantum modular forms. I will explain how, by introducing defects, a structure of vector-valued quantum modular forms emerges for families of defect invariants. These furthermore form false-mock pairs, where the proposed mock invariants are regularised indefinite theta-functions. In this form, similarly to the false invariants, the mock counterparts can also be related to corresponding vertex operator algebras.