arXiv subjects

Claudia Rella

Publications and source records attributed to Claudia Rella.

8 recordsLinked to original sources

arXiv2026

Modular resurgent structures for vectors

Building on prior results [1], we introduce vector-valued modular resurgent series, whose components exhibit a single infinite tower of singularities in the Borel plane, trivial secondary resurgent series, and Stokes constants given by linear combinations of the coefficients of a vector of $L$-functions. We extend the paradigm of modular resurgence to this setting, emphasizing the role of the Stokes constants and the interplay between the associated vectors of $q$-series and Dirichlet series, and describing the resulting symmetry relating canonical pairs of vector-valued modular resurgent series. Moreover, we conjecture that certain vectors of $q$-series with modular resurgent asymptotics are vector-valued quantum modular forms and can be reconstructed via median resummation. Finally, we show that vectors of $q$-Pochhammer symbols, previously considered in [2], and Eichler integrals of vector-valued modular forms of weight $1/2$ and $3/2$ can be studied within the framework of vector-valued modular resurgence. While the first case amounts to a convenient repackaging of scalar modular resurgence, the second involves a non-trivial representation of the modular group and therefore illustrates the necessity of the vector-valued framework; we establish its modular resurgent structure in general, and work it out in full detail for the unary theta series, whose Eichler integrals are the false theta functions of quantum topology.

math.NT↗

arXiv2025

Modular resurgence, $q$-Pochhammer symbols, and quantum operators from mirror curves

Building on the results of [1,2], we study the resurgence of $q$-Pochhammer symbols and determine their summability and quantum modularity properties. We construct a new, infinite family of pairs of modular resurgent series from the asymptotic expansions of sums of $q$-Pochhammer symbols weighted by suitable Dirichlet characters. These weighted sums fit into the modular resurgence paradigm and provide further evidence supporting our conjectures in [1]. In the context of the topological string/spectral theory correspondence for toric Calabi-Yau threefolds, Kashaev and Mari\~no proved that the spectral traces of canonical quantum operators associated with local weighted projective planes can be expressed as sums of $q$-Pochhammer symbols. Exploiting this relation, we show that an exact strong-weak resurgent symmetry, first observed by the second author in [3] and fully formalized in [2] for local $\mathbb{P}^2$, applies to all local $\mathbb{P}^{m,n}$, albeit stripped of some of the underlying number-theoretic properties. Under some assumptions, these properties are restored when considering linear combinations of the spectral traces that reproduce the weighted sums above.

hep-th↗

arXiv2024

Modular resurgent structures

The theory of resurgence uniquely associates a factorially divergent formal power series with a collection of exponentially small non-perturbative corrections paired with a set of complex numbers known as Stokes constants. When the Borel plane displays a single infinite tower of singularities, the secondary resurgent series are trivial, and the Stokes constants are coefficients of an $L$-function, a rich analytic number-theoretic fabric underlies the resurgent structure of the asymptotic series. We propose a new paradigm of modular resurgence that focuses on the role of the Stokes constants and the interplay of the $q$-series acting as their generating functions with the corresponding $L$-functions. Guided by two pivotal examples arising from topological string theory and the theory of Maass cusp forms, we introduce the notion of modular resurgent series, which we conjecture to have specific summability properties as well as to be related to quantum modular forms.

math.NT↗

arXiv2024

Strong-weak symmetry and quantum modularity of resurgent topological strings on local $\mathbb{P}^2$

Quantizing the mirror curve to a toric Calabi-Yau threefold gives rise to quantum operators whose fermionic spectral traces produce factorially divergent formal power series in the Planck constant and its inverse. These are conjecturally captured by the Nekrasov-Shatashvili and standard topological string free energies, respectively, via the TS/ST correspondence. The resurgent structures of the first fermionic spectral trace of local $\mathbb{P}^2$ in both weak and strong coupling limits were solved exactly by the second author in [1]. Here, we argue that a full-fledged strong-weak resurgent symmetry is at play, exchanging the perturbative/non-perturbative contributions to the holomorphic and anti-holomorphic blocks in the factorization of the spectral trace. This relies on a global net of relations connecting the perturbative series and the discontinuities in the dual regimes, which is built upon the analytic properties of the $L$-functions with coefficients given by the Stokes constants and the $q$-series acting as their generating functions. Then, we show that the latter are holomorphic quantum modular forms for $\Gamma_1(3)$ and are reconstructed by the median resummation of their asymptotic expansions.

hep-th↗

arXiv2024

Searching for Muonphilic Dark Sectors with Proton Beams

Proton beam-dump experiments are a high-intensity source of secondary muons and provide an opportunity to probe muon-specific dark sectors. We adopt a simplified-models framework for an exotic light scalar particle coupling predominantly or exclusively to muons. Equipped with state-of-the-art muon simulations, we compute the sensitivity reach in the parameter space $(m_S, \, g_μ)$ of the dark mediator, examining in detail the examples of the experiment NA62 in beam-dump mode and the proposed experiment SHiP. We find a significant yield of such exotics in the sub-GeV mass range. Our projections are competitive with those of primary muon-beam experiments and complementary to current constraints, spanning uncharted parameter space and accessing new physics potentially responsible for the $(g-2)_μ$ anomaly.

hep-ph↗

arXiv2023

On the Structure of Wave Functions in Complex Chern-Simons Theory

We study the structure of wave functions in complex Chern-Simons theory on the complement of a hyperbolic knot, emphasizing the similarities with the topological string/spectral theory correspondence. We first conjecture a hidden integrality structure in the holomorphic blocks and show that this structure guarantees the cancellation of potential singularities in the full non-perturbative wave function at rational values of the coupling constant. We then develop various techniques to determine the wave function at such rational points. Finally, we illustrate our conjectures and obtain explicit results in the examples of the figure-eight and three-twist knots. In the case of the figure-eight knot, we also perform a direct evaluation of the state integral in the rational case and observe that the resulting wave function has the features of the ground state for a quantum mirror curve.

hep-th↗

arXiv2023

Resurgence, Stokes constants, and arithmetic functions in topological string theory

The quantization of the mirror curve to a toric Calabi-Yau threefold gives rise to quantum-mechanical operators, whose fermionic spectral traces produce factorially divergent power series in the Planck constant. These asymptotic expansions can be promoted to resurgent trans-series. They show infinite towers of periodic singularities in their Borel plane and infinitely many rational Stokes constants, which are encoded in generating functions expressed in closed form in terms of $q$-series. We provide an exact solution to the resurgent structure of the first fermionic spectral trace of the local $\mathbb{P}^2$ geometry in the semiclassical limit of the spectral theory, corresponding to the strongly-coupled regime of topological string theory on the same background in the conjectural TS/ST correspondence. Our approach straightforwardly applies to the dual weakly-coupled limit of the topological string. We present and prove closed formulae for the Stokes constants as explicit arithmetic functions and for the perturbative coefficients as special values of known $L$-functions, while the duality between the two scaling regimes of strong and weak string coupling constant appears in number-theoretic form. A preliminary numerical investigation of the local $\mathbb{F}_0$ geometry unveils a more complicated resurgent structure with logarithmic sub-leading asymptotics. Finally, we obtain a new analytic prediction on the asymptotic behavior of the fermionic spectral traces in an appropriate WKB double-scaling regime, which is captured by the refined topological string in the Nekrasov-Shatashvili limit.

hep-th↗

arXiv2021

An Introduction to Motivic Feynman Integrals

This article gives a short step-by-step introduction to the representation of parametric Feynman integrals in scalar perturbative quantum field theory as periods of motives. The application of motivic Galois theory to the algebro-geometric and categorical structures underlying Feynman graphs is reviewed up to the current state of research. The example of primitive log-divergent Feynman graphs in scalar massless $ϕ^4$ quantum field theory is analysed in detail.

math-ph↗