Resurgent-transseries solutions to Painlevé equations may be recursively constructed out of these nonlinear differential-equations—but require Stokes data to be globally defined over the complex plane. Stokes data explicitly construct connection-formulae which describe the nonlinear Stokes phenomena associated to these solutions, via implementation of Stokes transitions acting on the transseries. Nonlinear resurgent Stokes data lack, however, a first-principle computational approach, hence are hard to determine generically. In the Painlevé I and Painlevé II contexts, nonlinear Stokes data get further hindered as these equations are resonant, with non-trivial consequences for the interconnections between transseries sectors, bridge equations, and associated Stokes coefficients. In parallel to this, the Painlevé I and Painlevé II equations are string-equations for two-dimensional quantum (super) gravity and minimal string theories, where Stokes data have natural ZZ-brane interpretations. This work conjectures for the first time the complete, analytical, resurgent Stokes data for the first two Painlevé equations, alongside their quantum gravity or minimal string incarnations. The method developed herein, dubbed “closed-form asymptotics”, makes sole use of resurgent large-order asymptotics of transseries solutions—alongside a careful analysis of the role resonance plays. Given its generality, it may be applicable to other distinct (nonlinear, resonant) problems. Results for analytical Stokes coefficients have natural structures, which are described, and extensive high-precision numerical tests corroborate all analytical predictions. Connection-formulae are explicitly constructed, with rather simple and compact final results encoding the full Stokes data, and further allowing for exact monodromy checks—hence for an analytical proof of our Painlevé I results.
The $c_2$-invariant is an arithmetic graph invariant useful for understanding Feynman periods. Brown and Schnetz conjectured that the $c_2$-invariant has a particular symmetry known as completion invariance. This paper will prove completion invariance of the $c_2$-invariant in the $p=2$ case, extending previous work of one of us. The methods are combinatorial and enumerative involving counting certain partitions of the edges of the graph.
We prove a conjecture of Bousseau, van Garrel and the first-named author relating, under suitable positivity conditions, the higher genus maximal contact $\log$ Gromov–Witten invariants of Looijenga pairs to other curve counting invariants of Gromov–Witten/Gopakumar–Vafa type. The proof consists of a closed-form $q$-hypergeometric resummation of the quantum tropical vertex calculation of the $\log$ invariants in presence of infinite scattering. The resulting identity of $q$-series appears to be new and of independent combinatorial interest.
We generalise a result of Kazarian regarding Kadomtsev–Petviashvili integrability for single Hodge integrals to general cohomological field theories related to Hurwitz-type counting problems or hypergeometric tau‑functions. The proof uses recent results on the relations between hypergeometric tau-functions and topological recursion, as well as the DOSS correspondence between topological recursion and cohomological field theories. As a particular case, we recover the result of Alexandrov of KP integrability for triple Hodge integrals with a Calabi–Yau condition.
We provide Fourier expansions of vector-valued eigenfunctions of the hyperbolic Laplacian that are twist-periodic in a horocycle direction. The twist may be given by any endomorphism of a finite-dimensional vector space; no assumptions on invertibility or unitarity are made. Examples of such eigenfunctions include vector-valued twisted automorphic forms of Fuchsian groups. We further provide a detailed description of the Fourier coefficients and explicitly identify each of their constituents, which intimately depend on the eigenvalues of the twisting endomorphism and the size of its Jordan blocks. In addition, we determine the growth properties of the Fourier coefficients.
We investigate Feynman graphs and their Feynman rules from the viewpoint of graph complexes. We focus on the interplay between graph homology, Hopf-algebraic structures on Feynman graphs and the analytic structure of their associated integrals. Furthermore, we discuss the appearance of cubical complexes where the differential is formed by reducing internal edges and by putting edge-propagators on the mass-shell.
We compute the fundamental group of an open Richardson variety in the manifold of complete flags that corresponds to a partial flag manifold. Rietsch showed that these $\log$ Calabi–Yau varieties underlie a Landau–Ginzburg mirror for the Langlands dual partial flag manifold, and our computation verifies a prediction of Hori for this mirror. It is $\log$ Calabi–Yau as it isomorphic to the complement of the Knutson–Lam–Speyer anti-canonical divisor for the partial flag manifold. We also determine explicit defining equations for this divisor.
We present basic constructions and properties in arithmetic Chern–Simons theory with finite gauge group along the line of topological quantum field theory. For a finite set $S$ of finite primes of a number field $k$, we construct arithmetic analogues of the Chern–Simons $1$-cocycle, the prequantization bundle for a surface and the Chern–Simons functional for a $3$-manifold. We then construct arithmetic analogues for $k$ and $S$ of the quantum Hilbert space (space of conformal blocks) and the Dijkgraaf–Witten partition function in $(2+1)$-dimensional Chern–Simons TQFT. We show some basic and functorial properties of those arithmetic analogues. Finally, we show decomposition and gluing formulas for arithmetic Chern–Simons invariants and arithmetic Dijkgraaf–Witten partition functions.