Tomás Reis | Searcharxiv

arXiv2025

On the Structure of Trans-Series in Quantum Field Theory

Many observables in quantum field theory can be expressed in terms of trans-series, in which one adds to the perturbative series a typically infinite sum of exponentially small corrections, due to instantons or to renormalons. Even after Borel resummation of the series in the coupling constant, one has to sum this infinite series of small exponential corrections. It has been argued that this leads to a new divergence, which is sometimes called the divergence of the OPE. We show that, in some interesting examples in quantum field theory, the series of small exponential corrections is convergent, order by order in the coupling constant. In particular, we give numerical evidence for this convergence property in the case of the free energy of integrable asymptotically free theories, which has been intensively studied recently in the framework of resurgence. Our results indicate that, in these examples, the Borel resummed trans-series leads to a well defined function, and there are no further divergences.

hep-th↗

arXiv2019

A new approximation method for geodesics on the space of Kähler metrics using complexified symplectomorphisms and Gröbner Lie series

It has been shown that the Cauchy problem for geodesics in the space of Kähler metrics with a fixed cohomology class on a compact complex manifold $M$ can be effectively reduced to the problem of finding the flow of a related hamiltonian vector field $X_H$, followed by analytic continuation of the time to complex time. This opens the possibility of expressing the geodesic $ω_t$ in terms of Gröbner Lie series of the form $\exp(\sqrt{-1} \, tX_H)(f)$, for local holomorphic functions $f$. The main goal of this paper is to use truncated Lie series as a new way of constructing approximate solutions to the geodesic equation. For the case of an elliptic curve and $H$ a certain Morse function squared, we approximate the relevant Lie series by their first twelve terms, calculated with the help of Mathematica. This leads to approximate geodesics which hit the boundary of the space of Kähler metrics in finite geodesic time. For quantum mechanical applications, one is interested also on the non-Kähler polarizations that one obtains by crossing the boundary of the space of Kähler structures. Properties of the approximate geodesics and its extensions are also studied using Mathematica.

math.SG↗