arXiv subjects

Tommaso Pedroni

Publications and source records attributed to Tommaso Pedroni.

3 recordsLinked to original sources

arXiv2026

Split Heun functions via blown-up surface defects

We study resonant solutions of the Heun equation and its confluent limits that arise in the Nekrasov-Shatashvili (NS) limit of four-dimensional $\mathcal{N}=2$ $\mathrm{SU}(2)$ gauge theories with fundamental hypermultiplets. At the resonant loci $2a/\hbar\in\mathbb{Z}$ in the Coulomb branch parameter $a$, the Floquet multipliers coalesce and the instanton expansions of the bulk and surface defect NS functions develop poles of increasing order. We derive blow-up equations involving exclusively NS functions and use them to resum these singular expansions. The resulting resummed bulk and surface defect NS functions reveal the analytic structure of the gauge-theoretic solutions near the resonant loci, including the branch structure of the accessory parameter and of the Floquet solutions that is obscured by the term-by-term instanton expansion. At resonance, the resummed accessory parameters and suitably normalized defect wavefunctions admit finite limits that describe periodic or antiperiodic solutions at the edges of spectral gaps and allow us to construct their logarithmic companions. We then identify distinct nested mass loci governing gap closure and semisimple resonant monodromy. On the larger locus the band-edge accessory parameters coalesce, while on the smaller locus two independent resonant (anti)periodic Floquet solutions survive. We develop the general resummation procedure for $N_f=(n_0,n_1)$ theories with $n_i\leq 2$ $(i=0,1)$, and demonstrate it explicitly for the $N_f=(1,1)$ theory.

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arXiv2026

Eigenfunctions of deformed Schrödinger equations

We study the spectral problems associated with the finite-difference operators $H_N = 2 \cosh(p) + V_N(x)$, where $V_N(x)$ is an arbitrary polynomial potential of degree $N$. These systems can be regarded as a solvable deformation of the standard Schrödinger operators $p^2 + V_N(x)$, and they arise naturally from the quantization of the Seiberg-Witten curve of four-dimensional, $\mathcal{N} = 2$, SU(N) supersymmetric Yang-Mills theory. Using the open topological string/spectral theory correspondence, we construct exact, generalized eigenfunctions of $H_N$, valid for arbitrary polynomial potentials and describing both bound and resonant states. We also comment on the case with a $\sinh(p)$ kinetic term. Our solutions are entire in $x$ for all generalized eigenvalues, and become square-integrable for a discrete subset of those. An interesting feature is the existence of special loci in the parameter space of the potential, where the eigenfunctions exhibit enhanced decay, leading to spectral degeneracies for confining potentials and to a real energy spectrum for unbounded ones. Our results provide a rare example of a quantum-mechanical spectral problem that is exactly solvable, admitting explicit, analytic eigenfunctions for both bound and resonant states.

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arXiv2025

Blowing-up the edge: connection formulae and stability chart of the Lamé equation

We study periodic spectral problems through their connection with supersymmetric gauge theories and two-dimensional conformal field theory. To characterize the associated stability chart, we develop a novel and systematic approach for analyzing semi-classical Virasoro blocks near their poles. Via the AGT correspondence, these blocks correspond to SU(2) Nekrasov partition functions in the Nekrasov-Shatashvili limit, which we propose to resum using an appropriate limit of blow-up equations. We show that the analytic structure of the resulting resummed partition functions features branch cuts located precisely at the edges between bands and gaps in the spectrum of the associated quantum integrable system with periodic potential. We examine the Nekrasov partition functions of $\mathcal{N}=2$ SQCD with $N_f \le 4$ flavors and of the $\mathcal{N}=2^*$ theory, which are related to the Heun equation, its confluent forms, and the Lamé equation. In the latter case, we analyze the spectrum in detail and solve the associated connection problem. Finally, we compare our results with those obtained via isomonodromic deformation techniques and the computation of orbifold surface defect partition functions in the $\mathcal{N}=2^*$ gauge theory, finding perfect agreement.

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