Resurgence and 1/N Expansion in Integrable Field Theories

In theories with renormalons the perturbative series is factorially divergent even after restricting to a given order in 1/N, making the 1/N expansion a natural testing ground for the theory of resurgence. We study in detail the interplay between resurgent properties and the 1/N expansion in various integrable field theories with renormalons. We focus on the free energy in the presence of a chemical potential coupled to a conserved charge, which can be computed exactly with the thermodynamic Bethe ansatz (TBA). In some examples, like the first 1/N correction to the free energy in the non-linear sigma model, the terms in the 1/N expansion can be fully decoded in terms of a resurgent trans-series in the coupling constant. In the principal chiral field we find a new, explicit solution for the large N free energy which can be written as the median resummation of a trans-series with infinitely many, analytically computable IR renormalon corrections. However, in other examples, like the Gross-Neveu model, each term in the 1/N expansion includes non-perturbative corrections which can not be predicted by a resurgent analysis of the corresponding perturbative series. We also study the properties of the series in 1/N. In the Gross-Neveu model, where this is convergent, we analytically continue the series beyond its radius of convergence and show how the continuation matches with known dualities with sine-Gordon theories.

Resurgent Analysis of Ward-Schwinger-Dyson Equations

Building on our recent derivation of the Ward-Schwinger-Dyson equations for the cubic interaction model, we present here the first steps of their resurgent analysis. In our derivation of the WSD equations, we made sure that they had the properties of compatibility with the renormalisation group equations and independence from a regularisation procedure which was known to allow for the comparable studies in the Wess-Zumino model. The interactions between the transseries terms for the anomalous dimensions of the field and the vertex is at the origin of unexpected features, for which the effect of higher order corrections is not precisely known at this stage: we are only at the beginning of the journey to use resurgent methods to decipher non-perturbative effects in quantum field theory.

Resurgent analysis for some 3-manifold invariants

We study resurgence for some 3-manifold invariants when Gℂ = SL(2, ℂ). We discuss the case of an infinite family of Seifert manifolds for general roots of unity and the case of the torus knot complement in S3. Via resurgent analysis, we see that the contribution from the abelian flat connections to the analytically continued Chern-Simons partition function contains the information of all non-abelian flat connections, so it can be regarded as a full partition function of the analytically continued Chern-Simons theory on 3-manifolds M3. In particular, this directly indicates that the homological block for the torus knot complement in S3 is an analytic continuation of the full G = SU(2) partition function, i.e. the colored Jones polynomial.

Resurgent analysis of SU(2) Chern-Simons partition function on Brieskorn spheres Σ(2, 3, 6n + 5)

$$ \hat{Z} $$ Z ̂ -invariants, which can reconstruct the analytic continuation of the SU(2) Chern-Simons partition functions via Borel resummation, were discovered by GPV and have been conjectured to be a new homological invariant of 3-manifolds which can shed light onto the superconformal and topologically twisted index of 3d $$ \mathcal{N} $$ N = 2 theories proposed by GPPV. In particular, the resurgent analysis of $$ \hat{Z} $$ Z ̂ has been fruitful in discovering analytic properties of the WRT invariants. The resurgent analysis of these $$ \hat{Z} $$ Z ̂ -invariants has been performed for the cases of Σ(2, 3, 5), Σ(2, 3, 7) by GMP, Σ(2, 5, 7) by Chun, and, more recently, some additional Seifert manifolds by Chung and Kucharski, independently. In this paper, we extend and generalize the resurgent analysis of $$ \hat{Z} $$ Z ̂ on a family of Brieskorn homology spheres Σ(2, 3, 6n + 5) where n ∈ ℤ+ and 6n + 5 is a prime. By deriving $$ \hat{Z} $$ Z ̂ for Σ(2, 3, 6n + 5) according to GPPV and Hikami, we provide a formula where one can quickly compute the non-perturbative contributions to the full analytic continuation of SU(2) Chern-Simons partition function.

On a Problem Arising in Application of the Re-Quantization Method to Construct Asymptotics of Solutions to Linear Differential Equations with Holomorphic Coefficients at Infinity

The re-quantization method—one of the resurgent analysis methods of current importance—is developed in this study. It is widely used in the analytical theory of linear differential equations. With the help of the re-quantization method, the problem of constructing the asymptotics of the inverse Laplace–Borel transform is solved for a particular type of functions with holomorphic coefficients that exponentially grow at zero. Two examples of constructing the uniform asymptotics at infinity for the second- and forth-order differential equations with the help of the re-quantization method and the result obtained in this study are considered.

The grin of Cheshire cat resurgence from supersymmetric localization

First we compute the \mbox{S}^2S2 partition function of the supersymmetric \mathbb{CP}^{N-1}ℂℙN−1 model via localization and as a check we show that the chiral ring structure can be correctly reproduced. For the \mathbb{CP}^1ℂℙ1 case we provide a concrete realisation of this ring in terms of Bessel functions. We consider a weak coupling expansion in each topological sector and write it as a finite number of perturbative corrections plus an infinite series of instanton-anti-instanton contributions. To be able to apply resurgent analysis we then consider a non-supersymmetric deformation of the localized model by introducing a small unbalance between the number of bosons and fermions. The perturbative expansion of the deformed model becomes asymptotic and we analyse it within the framework of resurgence theory. Although the perturbative series truncates when we send the deformation parameter to zero we can still reconstruct non-perturbative physics out of the perturbative data in a nice example of Cheshire cat resurgence in quantum field theory. We also show that the same type of resurgence takes place when we consider an analytic continuation in the number of chiral fields from NN to r\in\mathbb{R}r∈ℝ. Although for generic real rr supersymmetry is still formally preserved, we find that the perturbative expansion of the supersymmetric partition function becomes asymptotic so that we can use resurgent analysis and only at the end take the limit of integer rr to recover the undeformed model.