Stokes Phenomenon Research Articles

Wave function of simple universes analytically continued from negative to positive potentials

We elaborate on the correspondence between the canonical partition function in asymptotically AdS universes and the no-boundary proposal for positive vacuum energy. For the case of a pure cosmological constant, the analytic continuation of the AdS partition function is seen to define the no-boundary wave function (in dS) uniquely in the simplest minisuperspace model. A consideration of the AdS gravitational path integral implies that on the dS side, saddle points with Hawking-Moss/Coleman-De Luccia-type tunnelling geometries are irrelevant. This implies that simple topology changing geometries do not contribute to the nucleation of the universe. The analytic AdS/dS equivalence holds up once tensor fluctuations are added. It also works, at the level of the saddle point approximation, when a scalar field with a mass term is included, though in the latter case, it is the mass that must be analytically continued. Our results illustrate the emergence of time from space by means of a Stokes phenomenon, in the case of positive vacuum energy. Furthermore, we arrive at a new characterisation of the no-boundary condition, namely that there should be no momentum flux at the nucleation of the universe.

Stokes phenomena in 3d mathcal{N} = 2 SQED2 and mathbbm{CP} 1 models

We propose a novel approach of uncovering Stokes phenomenon exhibited by the holomorphic blocks of mathbbm{CP} 1 model by considering it as a specific decoupling limit of SQED2 model. This approach involves using a ℤ3 symmetry that leaves the supersymmetric parameter space of SQED2 model invariant to transform a pair of SQED2 holomorphic blocks to get two new pairs of blocks. The original pair obtained by solving the line operator identities of the SQED2 model and the two new transformed pairs turn out to be related by Stokes-like matrices. These three pairs of holomorphic blocks can be reduced to the known triplet of mathbbm{CP} 1 blocks in a particular decoupling limit where two of the chiral multiplets in the SQED2 model are made infinitely massive. This reduction then correctly reproduces the Stokes regions and matrices of the mathbbm{CP} 1 blocks. Along the way, we find six pairs of SQED2 holomorphic blocks in total, which lead to six Stokes-like regions covering uniquely the full parameter space of the SQED2 model.

Superadiabatic basis in cosmological particle production: application to preheating

We discuss the adiabatic basis dependence of particle number in time-dependent backgrounds. In particular, we focus on preheating after inflation, and show that, for the optimal basis, the time dependence of the produced particle number can be well approximated by a simple connection formula, which can be obtained by analysing Stokes phenomenon in given backgrounds. As we show explicitly, the simple connection formula can describe various parameter regions such as narrow and broad resonance regime in a unified manner.

Quantum phase transition and resurgence: Lessons from three-dimensional $\mathcal{N}=4$ supersymmetric quantum electrodynamics

Abstract We study a resurgence structure of a quantum field theory with a phase transition to uncover relations between resurgence and phase transitions. In particular, we focus on three-dimensional $\mathcal{N}=4$ supersymmetric quantum electrodynamics (SQED) with multiple hypermultiplets, where a second-order quantum phase transition has recently been proposed in the large-flavor limit. We provide interpretations of the phase transition from the viewpoints of Lefschetz thimbles and resurgence. For this purpose, we study the Lefschetz thimble structure and properties of the large-flavor expansion for the partition function obtained by the supersymmetric localization. We show that the second-order phase transition is understood as a phenomenon where a Stokes and an anti-Stokes phenomenon occur simultaneously. The order of the phase transition is determined by how saddles collide at the critical point. In addition, the phase transition accompanies an infinite number of Stokes phenomena due to the supersymmetry. These features are appropriately mapped to the Borel plane structures as the resurgence theory expects. Given the lessons from SQED, we provide a more general discussion on the relationship between the resurgence and phase transitions. In particular, we show how the information on the phase transition is decoded from the Borel resummation technique.

Stokes phenomenon and gravitational particle production — How to evaluate it in practice

We revisit gravitational particle production from the Stokes phenomenon viewpoint, which helps us make a systematic way to understand asymptotic behavior of mode functions in time-dependent background. One of our purposes of this work is to make the method more practical for evaluation of non-perturbative particle production rate. In particular, with several examples of time-dependent backgrounds, we introduce some approximation methods that make the analysis more practical. Specifically, we consider particle production in simple expanding backgrounds, preheating after R2 inflation, and a transition model with smoothly changing mass. As we find several technical issues in analyzing the Stokes phenomenon of each example, we discuss how to simplify the problems while showing the accuracy of analytic estimation under the approximations we make.

D-modules of pure Gaussian type and enhanced ind-sheaves

Differential systems of pure Gaussian type are examples of D-modules on the complex projective line with an irregular singularity at infinity, and as such are subject to the Stokes phenomenon. We employ the theory of enhanced ind-sheaves and the Riemann–Hilbert correspondence for holonomic D-modules of D’Agnolo and Kashiwara to describe the Stokes phenomenon topologically. Using this description, we perform a topological computation of the Fourier–Laplace transform of a D-module of pure Gaussian type in this framework, recovering and generalizing a result of Sabbah.

Transseries for causal diffusive systems

The large proper-time behaviour of expanding boost-invariant fluids has provided many crucial insights into quark-gluon plasma dynamics. Here we formulate and explore the late-time behaviour of nonequilibrium dynamics at the level of linearized perturbations of equilibrium, but without any special symmetry assumptions. We introduce a useful quantitative approximation scheme in which hydrodynamic modes appear as perturbative contributions while transients are nonperturbative. In this way, solutions are naturally organized into transseries as they are in the case of boost-invariant flows. We focus our attention on the ubiquitous telegrapher’s equation, the simplest example of a causal theory with a hydrodynamic sector. In position space we uncover novel transient contributions as well as Stokes phenomena which change the structure of the transseries based on the spacetime region or the choice of initial data.

The exact WKB for cosmological particle production

The Bogoliubov transformation in cosmological particle production can be explained by the Stokes phenomena of the corresponding ordinary differential equation. The calculation becomes very simple as far as the solution is described by a special function. Otherwise, the calculation requires more tactics, where the Exact WKB (EWKB) may be a powerful tool. Using the EWKB, we discuss cosmological particle production focusing on the effect of more general interaction and classical scattering. The classical scattering appears when the corresponding scattering problem of the Schrödinger equation develops classical turning points on the trajectory. The higher process of fermionic preheating is also discussed using the Landau-Zener model.

Particle production induced by vacuum decay in real time dynamics

We discuss particle production associated with vacuum decay, which changes the mass of a scalar field coupled to a background field which induces the decay. By utilizing the Stokes phenomenon, we can optimally track the time-evolution of mode function and hence calculate particle production properly. In particular, we use real time formalisms for vacuum decay in Minkowski and de Sitter spacetime together with the Stokes phenomenon method. For each case, we consider the flyover vacuum decay model and stochastic inflation, respectively. Within the real time formalism, the particle production can be viewed as that caused by nontrivial external fields. This gives us a novel perspective of the real time formalism of vacuum decay.

Stokes phenomenon and Hawking radiation

We compute the semiclassical decay rate for Kerr black hole by deriving a one-way connection formula, relating the near horizon solution to the outgoing solution at infinity. In particular, we discuss the relevance of the Stokes phenomenon and show how it leads to a Boltzmann-like thermal weight factor by making use of the Stokes diagrams. We also give the exact result for the semiclassical greybody factor $e^{-2S}, $ where $S$ is the leading order WKB action. We contrast our results with Maldacena and Strominger's work [Phys. Rev. D 56, 4975 (1997)], where the emission spectrum for a rotating black hole was computed locally via asymptotic matching. We find that the relative error of semiclassical decay rate with respect to asymptotic matching formula diminishes in the limit of large angular momentum, $l$, as expected. In this limit, the action assumes a compact form: $2S \sim (2 l+1) (\text{Log} \left( \frac{16 }{z_0}\right)-1)$, where $z_0$ is the cross ratio formed by the critical points (zeroes) of the scattering potential.

On exact-WKB analysis, resurgent structure, and quantization conditions

There are two well-known approaches to studying nonperturbative aspects of quantum mechanical systems: saddle point analysis of the partition functions in Euclidean path integral formulation and the exact-WKB analysis based on the wave functions in the Schrödinger equation. In this work, based on the quantization conditions obtained from the exact-WKB method, we determine the relations between the two formalism and in particular show how the two Stokes phenomena are connected to each other: the Stokes phenomenon leading to the ambiguous contribution of different sectors of the path integral formulation corresponds to the change of the “topology” of the Stoke curves in the exact-WKB analysis. We also clarify the equivalence of different quantization conditions including Bohr-Sommerfeld, path integral and Gutzwiller’s ones. In particular, by reorganizing the exact quantization condition, we improve Gutzwiller’s analysis in a crucial way by bion contributions (incorporating complex periodic paths) and turn it into an exact result. Furthermore, we argue the novel meaning of quasi-moduli integral and provide a relation between the Maslov index and the intersection number of Lefschetz thimbles.

A note on the Stokes phenomenon in flow under an elastic sheet.

The Stokes phenomenon is a class of asymptotic behaviour that was first discovered by Stokes in his study of the Airy function. It has since been shown that the Stokes phenomenon plays a significant role in the behaviour of surface waves on flows past submerged obstacles. A detailed review of recent research in this area is presented, which outlines the role that the Stokes phenomenon plays in a wide range of free surface flow geometries. The problem of inviscid, irrotational, incompressible flow past a submerged step under a thin elastic sheet is then considered. It is shown that the method for computing this wave behaviour is extremely similar to previous work on computing the behaviour of capillary waves. Exponential asymptotics are used to show that free-surface waves appear on the surface of the flow, caused by singular fluid behaviour in the neighbourhood of the base and top of the step. The amplitude of these waves is computed and compared to numerical simulations, showing excellent agreements between the asymptotic theory and computational solutions. This article is part of the theme issue 'Stokes at 200 (part 2)'.

Phase transition in complex-time Loschmidt echo of short and long range spin chain

We explain and exploit the random matrix formulation of the Loschmidt echo for the XX spin chain, valid for multiple domain wall initial states and also for an XX spin chain generalized with additional interactions to more neighbours. For models with interactions decaying as , with p integer or natural number and α ⩾ 0, we show that there are third order phase transitions in a scaling limit of the complex-time Loschmidt echo amplitudes. For the long-range version of the chain, we use an exact result for Toeplitz determinants with a pure Fisher–Hartwig singularity, to obtain exactly the Loschmidt echo for complex times and discuss the associated Stokes phenomena. We also study the case of a finite chain for one of the generalized XX models.

Topological computation of some Stokes phenomena on the affine line

Let $\mathcal M$ be a holonomic algebraic $\mathcal D$-module on the affine line, regular everywhere including at infinity. Malgrange gave a complete description of the Fourier-Laplace transform $\widehat{\mathcal M}$, including its Stokes multipliers at infinity, in terms of the quiver of $\mathcal M$. Let $F$ be the perverse sheaf of holomorphic solutions to $\mathcal M$. By the irregular Riemann-Hilbert correspondence, $\widehat{\mathcal M}$ is determined by the enhanced Fourier-Sato transform $F^\curlywedge$ of $F$. Our aim here is to recover Malgrange's result in a purely topological way, by computing $F^\curlywedge$ using Borel-Moore cycles. In this paper, we also consider some irregular $\mathcal M$'s, like in the case of the Airy equation, where our cycles are related to steepest descent paths.

Spin Effect and Stokes Phenomenon for Fermion Production in Electric Fields

One of the intriguing features of (Sauter-)Schwinger pair production is the Stokes phenomenon of no-pair production in some bipolar electric fields. The spin effect is studied on fermion pair production in the same electric fields of New Physics (Sae Mulli) 68, 1225 (2018), in which pair production of bosons is explained in terms of polons, simple poles with complex frequency. The Stokes phenomenon is showed to occur for specific parameters when the electric field is bipolar and changes the direction of antisymmetry in time. We compare the pair production of bosons and fermions and discuss its physical implications.

Reconstructing GKZ via Topological Recursion

In this article, a novel description of the hypergeometric differential equation found from Gel’fand–Kapranov–Zelevinsky’s system (referred to as GKZ equation) for Givental’s J-function in the Gromov–Witten theory will be proposed. The GKZ equation involves a parameter $$\hbar $$, and we will reconstruct it as a quantum curve from the classical limit $$\hbar \rightarrow 0$$ via the topological recursion. In this analysis, the spectral curve (referred to as GKZ curve) plays a central role, and it can be described by the critical point set of the mirror Landau–Ginzburg potential. Our novel description is derived via the duality relations of the string theories, and various physical interpretations suggest that the GKZ equation is identified with the quantum curve for the brane partition function in the cohomological limit. As an application of our novel picture for the GKZ equation, we will discuss the Stokes phenomenon for the equivariant $${\mathbb {C}}\mathbf{P }^{1}$$ model, and the wall-crossing formula for the total Stokes matrix will be examined. And as a byproduct of this analysis, we will study Dubrovin’s conjecture for this equivariant model.

Stokes Phenomenon and Yang–Baxter Equations

We describe the monodromy of dynamical Knizhnik-Zamolodchikov equations via Stokes phenomenon. It defines a family of braid groups representations by certain Stokes matrices. In particular, these Stokes matrices satisfy the Yang-Baxter equation.

Isomonodromy deformations at an irregular singularity with coalescing eigenvalues

We consider an $n\times n$ linear system of ODEs with an irregular singularity of Poincar\'e rank 1 at $z=\infty$, holomorphically depending on parameter $t$ within a polydisc in $\mathbb{C}^n$ centred at $t=0$. The eigenvalues of the leading matrix at $z=\infty$ coalesce along a locus $\Delta$ contained in the polydisc, passing through $t=0$. Namely, $z=\infty$ is a resonant irregular singularity for $t\in \Delta$. We analyse the case when the leading matrix remains diagonalisable at $\Delta$. We discuss the existence of fundamental matrix solutions, their asymptotics, Stokes phenomenon and monodromy data as $t$ varies in the polydisc, and their limits for $t$ tending to points of $\Delta$. When the deformation is isomonodromic away from $\Delta$, it is well known that a fundamental matrix solution has singularities at $\Delta$. When the system also has a Fuchsian singularity at $z=0$, we show under minimal vanishing conditions on the residue matrix at $z=0$ that isomonodromic deformations can be extended to the whole polydisc, including $\Delta$, in such a way that the fundamental matrix solutions and the constant monodromy data are well defined in the whole polydisc. These data can be computed just by considering the system at fixed $t=0$. Conversely, if the $t$-dependent system is isomonodromic in a small domain contained in the polydisc not intersecting $\Delta$, if the entries of the Stokes matrices with indices corresponding to coalescing eigenvalues vanish, then we show that $\Delta$ is not a branching locus for the fundamental matrix solutions. The importance of these results for the analytic theory of Frobenius Manifolds is explained. An application to Painlev\'e equations is discussed.

The q-Borel Sum of Divergent Basic Hypergeometric Series rφs(a;b;q,x)

We study the divergent basic hypergeometric series which is a $q$-analog of divergent hypergeometric series. This series formally satisfies the linear $q$-difference equation. In this paper, for that equation, we give an actual solution which admits basic hypergeometric series as a $q$-Gevrey asymptotic expansion. Such an actual solution is obtained by using $q$-Borel summability, which is a $q$-analog of Borel summability. Our result shows a $q$-analog of the Stokes phenomenon. Additionally, we show that letting $q\to1$ in our result gives the Borel sum of classical hypergeometric series. The same problem was already considered by Dreyfus, but we note that our result is remarkably different from his one.

Non-perturbative string theory from AdS/CFT

The large N expansion of giant graviton correlators is considered. Giant gravitons are described using operators with a bare dimension of order N . In this case the usual 1/N expansion is not applicable and there are contributions to the correlator that are non-perturbative in character. By writing the (square of the) correlators in terms of the hypergeometric function 2F1(a, b; c; 1), we are able to rephrase the 1/N expansion of the correlator as a semi-classical expansion for a Schrödinger equation. In this way we are able to argue that the 1/N expansion of the correlator is Borel summable and that it exhibits a parametric Stokes phenomenon as the angular momentum of the giant graviton is varied.