The computation of observables in general interacting theories, be them quantum mechanical, field, gauge or string theories, is a non-trivial problem which in many cases can only be addressed by resorting to perturbative methods. In most physically interesting problems these perturbative expansions result in asymptotic series with zero radius of convergence. These asymptotic series then require the use of resurgence and transseries in order for the associated observables to become nonperturbatively well-defined. Resurgence encodes the complete largeorder asymptotic behaviour of the coefficients from a perturbative expansion, generically in terms of (multi) instanton sectors and for each problem in terms of its Stokes constants. Some observables arise from linear problems, and have a finite number of instanton sectors and associated Stokes constants; some other observables arise from nonlinear problems, and have an infinite number of instanton sectors and Stokes constants. By means of two very explicit examples, and with emphasis on a pedagogical style of presentation, this work aims at serving as a primer on the aforementioned resurgent, large-order asymptotics of general perturbative expansions. This includes discussions of transseries, Stokes phenomena, generalized steepest-descent methods, Borel transforms, nonlinear resonance, and alien calculus. Furthermore, resurgent properties of transseries-usually described mathematically via alien calculus-are recast in equivalent physical languages: either a "statistical mechanical" language, as motions in chains and lattices; or a "conformal field theoretical" language, with underlying Virasoro-like algebraic structures.

Weyl semimetals possess massless chiral quasi-particles, and are thus affected by the triangle anomalies. We discuss the features of the chiral magnetic and chiral vortical effects specific to Weyl semimetals, and then propose three novel phenomena caused by the triangle anomalies in this material: 1) anomaly cooling; 2) charge transport by soliton waves as described by the Burgers' equation, and 3) the shift of the BKT phase transition of superfluid vortices coupled to Weyl fermions. In addition, we establish the conditions under which the chiral magnetic current exists in real materials.

We introduce a novel photon production mechanism stemming from the conformal anomaly of QCD×QED and the existence of strong (electro)magnetic fields in heavy ion collisions. Using the hydrodynamical description of the bulk modes of QCD plasma, we show that this mechanism leads to the photon production yield that is comparable to the yield from conventional sources. This mechanism also provides a significant positive contribution to the azimuthal anisotropy of photons, v(2), as well as to the radial "flow." We compare our results to the data from the PHENIX Collaboration.

A possible solution of the notorious sign problem preventing direct Monte Carlo calculations for systems with non-zero chemical potential is to deform the integration region in the complex plane to a Lefschetz thimble. We investigate this approach for a simple fermionic model. We introduce an easy to implement Monte Carlo algorithm to sample the dominant thimble. Our algorithm relies only on the integration of the gradient flow in the numerically stable direction, which gives it a distinct advantage over the other proposed algorithms. We demonstrate the stability and e ciency of the algorithm by applying it to an exactly solvable fermionic model and compare our results with the analytical ones. We report a very good agreement for a certain region in the parameter space where the dominant contribution comes from a single thimble, including a region where standard methods su↵er from a severe sign problem. However, we find that there are also regions in the parameter space where the contribution from multiple thimbles is important, even in the continuum limit.2

The Nekrasov-Shatashvili limit for the low-energy behavior of N = 2 and N = 2 * supersymmetric SU (2) gauge theories is encoded in the spectrum of the Mathieu and Lamé equations, respectively. This correspondence is usually expressed via an all-orders Bohr-Sommerfeld relation, but this neglects non-perturbative effects, the nature of which is very different in the electric, magnetic and dyonic regions. In the gauge theory dyonic region the spectral expansions are divergent, and indeed are not Borel-summable, so they are more properly described by resurgent trans-series in which perturbative and non-perturbative effects are deeply entwined. In the gauge theory electric region the spectral expansions are convergent, but nevertheless there are non-perturbative effects due to poles in the expansion coefficients, and which we associate with worldline instantons. This provides a concrete analog of a phenomenon found recently by Drukker, Mariño and Putrov in the large N expansion of the ABJM matrix model, in which non-perturbative effects are related to complex spacetime instantons. In this paper we study how these very different regimes arise from an exact WKB analysis, and join smoothly through the magnetic region. This approach also leads to a simple proof of a resurgence relation found recently by Dunne andÜnsal, showing that for these spectral systems all non-perturbative effects are subtly encoded in perturbation theory, and identifies this with the Picard-Fuchs equation for the quantized elliptic curve.Keywords: N = 2 SU(2) theory, Nekrasov-Shatashvili limit, exact quantization, resurgence, allorders WKB, worldline instantons 10 The paper [35] uses another form of the elliptic potential, sd 2 (z, k 2 ), which has a manifest symmetry under k 2 → 1 − k 2 . This potential is related to the sn 2 (z, k 2 ) by a simple Landen transformation.

We point out that Monte Carlo simulations of theories with severe sign problems can be profitably performed over manifolds in complex space different from the one with fixed imaginary part of the action ("Lefschetz thimble"). We describe a family of such manifolds that interpolate between the tangent space at one critical point (where the sign problem is milder compared to the real plane but in some cases still severe) and the union of relevant thimbles (where the sign problem is mild but a multimodal distribution function complicates the Monte Carlo sampling). We exemplify this approach using a simple 0+1 dimensional fermion model previously used on sign problem studies and show that it can solve the model for some parameter values where a solution using Lefschetz thimbles was elusive.

We revisit the problem of dipole-dipole scattering via exchanges of soft Pomerons in the context of holographic QCD. We show that a single closed string exchange contribution to the eikonalized dipole-dipole scattering amplitude yields a Regge behavior of the elastic amplitude; the corresponding slope and intercept are different from previous results obtained by a variational analysis of semi-classical surfaces. We provide a physical interpretation of the semi-classical worldsheets driving the Regge behavior for (−t) > 0 in terms of worldsheet instantons. The latter describe the Schwinger mechanism for string pair creation by an electric field, where the longitudinal electric field EL = σT tanh(χ/2) at the origin of this non-perturbative mechanism is induced by the relative rapidity χ of the scattering dipoles. Our analysis naturally explains the diffusion in the impact parameter space encoded in the Pomeron exchange; in our picture, it is due to the Unruh temperature of accelerated strings under the electric field. We also argue for the existence of a "micro-fireball" in the middle of the transverse space due to the soft Pomeron exchange, which may be at the origin of the thermal character of multiparticle production in ep/pp collisions. After summing over uncorrelated multi-Pomeron exchanges, we find that the total dipole-dipole cross section obeys the Froissart unitarity bound.

A general quantum mechanical or quantum field theoretical system in the path integral formulation has both real and complex saddles (instantons and ghost-instantons). Resurgent asymptotic analysis implies that both types of saddles contribute to physical observables, even if the complex saddles are not on the integration path i.e., the associated Stokes multipliers are zero. We show explicitly that instanton-anti-instanton and ghost-antighost saddles both affect the expansion around the perturbative vacuum. We study a self-dual model in which the analytic continuation of the partition function to negative values of coupling constant gives a pathological exponential growth, but a homotopically independent combination of integration cycles (Lefschetz thimbles) results in a sensible theory. These two choices of the integration cycles are tied with a quantum phase transition. The general set of ideas in our construction may provide new insights into non-perturbative QFT, string theory, quantum gravity, and the theory of quantum phase transitions.