Branch-Point Singularities in Complex Time

• Branch-point singularities in complex time are points where analytic continuations yield multi-valued solutions, often described by Puiseux expansions with fractional powers.
• They play a crucial role in Hamiltonian systems, quantum transitions, and cosmological correlators by delineating integrability and guiding regularization methods such as blow-ups and Padé approximations.
• These singularities influence computations in differential equations and dynamical systems by defining convergence limits of series expansions and governing transitions near exceptional points.

Branch-point singularities in complex time are critical loci where the analytic structure of time-evolved solutions (or parameter-dependent spectra) becomes non-single-valued due to the emergence of multi-sheeted Riemann surfaces. These singularities arise in various domains of mathematical physics, dynamical systems, quantum mechanics, field theory, minimal surfaces, and cosmology, and they are often characterized by fractional or logarithmic multivaluedness in the analytic continuation of time-dependent objects such as solutions of differential equations, spectra, or correlators.

1. Foundational Definitions and General Properties

A branch-point singularity in complex time occurs at a value $t_0 \in \mathbb{C}$ where analytic continuation of a solution $y(t)$ (or another time-dependent object) yields multivaluedness, typically represented by a local Puiseux expansion: $y(t) = \sum_{k=0}^\infty c_k (t-t_0)^{k/n}$ with $n > 1$ (algebraic branch point), or in more complicated cases, involving logarithmic terms. The singularity is called movable if its location depends on initial conditions (not on fixed coefficients of the system); this is typical in nonlinear ODEs and Hamiltonian systems (Kecker, 2013, Kecker et al., 2021).

Algebraic (fractional-power) branch points are common in integrable and near-integrable systems with the Painlevé property or its generalizations (quasi-Painlevé property), indicating that all movable singularities are isolated and algebraic rather than essential or logarithmic (Kecker, 2013, Kecker et al., 2021).

2. Hamiltonian Systems and Painlevé Analysis

Polynomial Hamiltonian systems in two variables

$$H(z, y_1, y_2) = y_1^{M+1} + y_2^{N+1} + \sum_{(i,j)\in I} Q_{ij}(z) y_1^i y_2^j$$

with associated Hamiltonian equations for $y_1(z)$ and $y_2(z)$ exhibit only algebraic branch-point singularities upon analytic continuation along rectifiable curves in the $z$ (complex time) plane. The local expansions at singularities are Puiseux series with denominator $(MN-1)$, ensuring finite branching and the absence of logarithmic terms (Kecker, 2013).

The method of regularizing transformations—compactification to the projective plane, followed by a cascade of blow-ups—yields a "space of initial values" where all solutions admit analytic continuation through all points, except possibly a set of exceptional divisors, and identifies precisely when a system exhibits only algebraic branch-point singularities (no movable logarithmic or essential singularities). This geometric framework generalizes the classic Painlevé property (Kecker et al., 2021).

Algorithmically, resonance conditions arising from recursion in the Puiseux series signal potential logarithmic terms—if obstructions are absent by construction or via the blow-up process, only algebraic singularities are present. This approach applies to both ordinary and Hamiltonian systems (Kecker, 2013, Kecker et al., 2021).

3. Minimal Surfaces and Multi-Valued Graphs

Minimal graphs and stationary varifolds with branch-point singularities are analyzed via frequency functions and asymptotic expansions. At (most) branch points, a two-valued minimal graph $u$ decays modulo its average to a unique nonzero two-valued cylindrical harmonic tangent function: $$\varphi^{(Z)}(X) = \{ \pm \operatorname{Re}(c (x_1 + i x_2)^{k/2}) \}$$ with $k \geq 3$ and $c \in \mathbb{C}^m \setminus \{0\}$; the homogeneity degree is at least $3/2$. The branch locus is countably $(n-2)$-rectifiable and, in favorable cases (homogeneity $3/2$), forms an embedded real analytic submanifold. These results enable a stratification of the singular set based on the structure of tangent functions and their symmetries (Krummel et al., 2021).

4. Branch Points in Analytic Structures of Waves and Operators

For fully nonlinear Stokes waves on deep water, conformal mapping techniques expose singularities in the complexified conformal variable (often interpreted as "complex time"). The only finite-plane singularities are square-root branch points aligned per horizontal period; their position $w=i v_c$ (in the conformal $w$-plane) controls the analytic structure. The Padé approximation identifies these branch points and their cuts with exponential convergence as the number of approximating poles increases (Dyachenko et al., 2015).

Mapping to the $\zeta = \tan(\frac{1}{2} w)$ plane confines the essential branch cut to a finite segment. In the limiting wave, as $v_c\to 0$, these nested square root singularities accumulate, giving rise to a $2/3$-power law singularity at the wave crest. The analytic continuation of the Stokes wave embodies an infinite hierarchy of Riemann sheets, each with paired square-root branch points, leading ultimately to nontrivial scaling exponents (Lushnikov, 2015).

Spectral branch points in non-Hermitian operators (e.g., the Bloch-Torrey operator) also appear at parameter values (often associated with complex time in evolution equations) where eigenvalues and eigenvectors coalesce, forming square-root branch points and Jordan blocks. These features govern the breakdown radius of perturbative expansions and determine the phase transition between different eigenmode localizations (Moutal et al., 2022).

5. Branch Points in Quantum Dynamics and Exceptional Points

In non-Hermitian quantum systems driven by time-dependent parameters (e.g., two-level atoms under chirped pulses), the analytic continuation of the adiabatic Hamiltonian into the complex time plane reveals transition points (TPs)—branch points in complex time where instantaneous eigenvalues coalesce. The coalescence of two such TPs marks the transition between monotonic rapid adiabatic passage and Rabi oscillations. These phenomena are intertwined with the concept of exceptional points (EPs) in parameter space, forming a geometric and topological connection between the analytic structure in time and underlying system degeneracies. Experimental scenarios for probing these transitions utilize chirped laser pulses and measurements of survival probabilities as functions of pulse parameters (Kapralova-Zdanska et al., 2019, Kapralova-Zdanska, 2021).

The complex time method evaluates nonadiabatic amplitudes via contour integration in the complex time plane, with integrand singularities (arising as Puiseux expansions) at TPs/branch points. The survival probabilities are dictated by interference or decay, depending on the arrangement of central TPs and their coalescence (Kapralova-Zdanska, 2021).

6. Branch Points in Cosmological Correlators

In cosmological settings, especially in the analysis of inflationary correlators at one-loop order, time integrations in the in-in formalism result in poles in total and partial energies, which are dressed into branch cuts through momentum integrations. A robust "off-shell total energy branch point" emerges, leading to logarithmic branch cuts involving ratios of comoving scales, even in the presence of dimensional/cutoff regularization. This behavior neither cancels under the cosmological KLN theorem (as might be expected from flat-space analogies) nor disappears in renormalized quantities, but repackages itself in a dilatation-invariant form (i.e., arguments of logs as ratios scaling covariantly under rescalings of all momenta or energies) (Bhowmick et al., 27 Mar 2025).

Diagrammatic rules provide a predictive algorithm: each partial energy pole involving loop momenta produces, upon integration, a log-branch cut with arguments given by the (partial) energy injection through a subgraph. These rules systematize the extraction of both pole and branch-cut singularities from complex time integrals in inflationary correlators, greatly simplifying the analytic structure analysis (Bhowmick et al., 27 Mar 2025).

7. Singularities and Exponential Estimates in Dynamical Systems

For real analytic vector fields extended to complex time, the maximal analytic strip width $|\operatorname{Im} t| \leq \eta$ is bounded by the position of the nearest singularity (often a branch point or a pole) of connecting orbits in the complex time plane. This $\eta$ controls exponentially small effects such as separatrix splitting under rapid forcing or discretization: $$\text{Splitting} \leq C_\eta \exp(-\eta/\varepsilon)$$ where $\varepsilon$ is the relevant small parameter. If a homoclinic or periodic orbit is complex entire (i.e., analytic throughout the entire complex plane), the splitting can be ultra-exponentially small—a regime currently unproved except in certain time-reversible cases with explicit solutions. The existence of such entire solutions (with no finite-time singularities in complex time) is an open problem, and finding one would imply the possibility of "ultra-invisible" chaotic behavior, essentially unobservable due to the extreme smallness of the effects (Fiedler, 2023).

8. Physical and Mathematical Significance

Branch-point singularities in complex time are central to:

• The classification and integrability of ODEs (e.g., via the Painlevé and quasi-Painlevé properties).
• The analytic structure and multi-valuedness of solutions to nonlinear PDEs (Stokes waves, minimal surfaces).
• The emergence and suppression of nonadiabatic transitions and phase switches in quantum dynamics, notably through the connection to exceptional points and Riemann surface topology.
• The breakdown point of perturbation theory and emergence of critical phenomena (exceptional/spectral branch points).
• The structure and computability of cosmological correlators, tying singularity positions in complex time to physical requirements like scale invariance and unitarity.
• The regulation of exponentially small dynamic phenomena, such as chaos and resonant driving, through the distance to singularities in complex time.

9. Representative Mathematical Forms and Algorithms

Domain	Branch Point Type	Local Expansion/Condition
ODE/Hamiltonian (e.g., Painlevé)	Algebraic movable (fractional power)	$y(z) = \sum_k c_k (z-z_0)^{k/q}$
Stokes waves	Square-root, nested, higher-order	$\rho(\chi) \sim (\chi-\chi_c)^{1/2}$
Non-Hermitian operator	Spectral: eigenvalue coalescence	$$\lambda_\pm(g) \simeq \lambda_0 \pm \sqrt{d'(g_0)(g-g_0)}$$
Time-dependent Hamiltonian	Puiseux at TPs (transition points)	$\delta(t) \sim (t-t_k)^{1/2}$ (TP at $t_k$)
Inverse function (e.g., Langevin)	Square-root (multi-valued inversion)	$\mathscr{L}^{-1}(z) \sim (z-z_1)^{1/2}$

Analytic continuation, Puiseux series analysis, regularizing blow-ups, Padé approximation, and contour integration in the complex plane are essential analytic and numerical tools for classifying and computing branch-point singularities.

10. Outlook and Open Problems

The fine structure and implications of branch-point singularities in complex time underpin advances in integrable systems, quantum dynamics, minimal surface theory, spectrum of non-Hermitian operators, and cosmological perturbation theory. Open challenges include the explicit construction (or proof of nonexistence) of complex entire connecting orbits between regular equilibria (Fiedler, 2023) and the systematic classification of higher-order or nested branch points in systems approaching criticality (e.g., limiting Stokes waves (Lushnikov, 2015)). There is expanding utility for algorithmic approaches that leverage diagrammatic and geometric insights into singularity structure, including the interplay with symmetries, conservation laws, and global analytic properties.

---

References:

• (Kecker, 2013) Polynomial Hamiltonian Systems with Movable Algebraic Singularities
• (Dyachenko et al., 2015) Branch cuts of Stokes wave on deep water. Part I: Numerical solution and Padé approximation
• (Lushnikov, 2015) Branch cuts of Stokes wave on deep water. Part II: Structure and location of branch points in infinite set of sheets of Riemann surface
• (Kecker et al., 2021) Regularising Transformations for Complex Differential Equations with Movable Algebraic Singularities
• (Krummel et al., 2021) Fine properties of branch point singularities: stationary two-valued graphs and stable minimal hypersurfaces near points of density $< 3$
• (Moutal et al., 2022) Spectral branch points of the Bloch-Torrey operator
• (Kapralova-Zdanska et al., 2019) Coalescence of two branch points in complex time marks the end of rapid adiabatic passage and the start of Rabi oscillations
• (Kapralova-Zdanska, 2021) Complex time method for quantum dynamics when an exceptional point is encircled in the parameter space
• (Bhowmick et al., 27 Mar 2025) Singularities in Cosmological Loop Correlators
• (Fiedler, 2023) Real chaos and complex time