Exact WKB Analysis & Resurgence

• Exact WKB analysis is a non-perturbative framework that resums divergent semiclassical expansions and systematically tracks Stokes phenomena across the complex plane.
• It encodes quantum monodromy through Voros symbols and employs cluster algebra mutations to capture the analytic and resurgent properties of differential equations.
• The approach applies to Schrödinger-type equations, providing precise quantization conditions, nonperturbative corrections, and deep insights into spectral theory.

Exact WKB analysis is a non-perturbative framework for studying linear differential and difference equations—most notably Schrödinger-type equations—by systematically resumming the divergent WKB expansions and tracking their analytic continuation across the complex plane. It rigorously incorporates Stokes phenomena, clarifies the structure of quantum monodromy data, and encodes the global analytic and resurgence properties of the system. The central objects of the formalism are the Voros symbols—exponentiated quantum periods—whose mutation under deformation, as well as their jump behavior across Stokes curves, is controlled by both the analytic topology of the underlying Riemann surface and, in many applications, by deep connections with cluster algebras.

1. Fundamentals of the Exact WKB Expansion and Stokes Geometry

At the heart of exact WKB analysis is the Schrödinger equation with a meromorphic potential,

$$\left(\frac{d^2}{dz^2} - \eta^2\,Q(z,\eta)\right)\psi(z,\eta)=0\ ,$$

where $Q(z,\eta) = Q_0(z) + \eta^{-1} Q_1(z) + \cdots$ and $\eta$ (often $1/\hbar$) is the large parameter. The formal WKB solution is given by

$$\psi(z,\eta) = \exp\left(\int^z S(z',\eta)\,dz'\right),$$

with

$S(z, \eta) = \sum_{n=-1}^\infty S_n(z)\, \eta^{-n}$

and $S_{-1}(z) = \pm \sqrt{Q_0(z)}$. The geometry is described by the quadratic differential $\phi = Q_0(z)\,dz^2$. Stokes curves are defined by

$$\mathrm{Im} \int^{z} \sqrt{Q_0(z)} dz = \text{constant},$$

and organize the complex plane into Stokes regions.

Turning points (zeros of $Q_0(z)$) and higher-order poles determine the topology of the associated Riemann surface; their arrangements, tracked through deformation of parameters, generate a Stokes graph whose combinatorics underlie the mutation theory central to exact WKB (Iwaki et al., 2014). The analytic WKB connection problem is fully controlled by these Stokes graphs.

2. Voros Symbols, Stokes Automorphisms, and Resurgence

The central analytic invariants are the Voros symbols, constructed from quantum periods along cycles $\gamma$ on the double cover $\hat\Sigma$ where $\sqrt{Q_0(z)}$ is single-valued. For a cycle $\gamma$ and a relative path $\beta$, one defines

$$V_{\gamma}(\eta) = \oint_{\gamma} S_{\text{odd}}(z,\eta) dz, \qquad W_\beta(\eta)=\int_{\beta} \left(S_{\text{odd}}(z,\eta) - \eta \sqrt{Q_0(z)}\right)dz,$$

with $S_{\text{odd}}$ built from the difference of the two formal Riccati roots. The (generally divergent) formal series for the Voros coefficients must be Borel resummed.

Borel sums of Voros symbols display Stokes jumps when crossing Stokes curves, encapsulated by the DDP (Delabaere–Dillinger–Pham) formulas: $$\mathcal{S}_{-}[e^{W_\beta}] = \mathcal{S}_{+}[e^{W_\beta}]\, (1+\mathcal{S}_{+}[e^{V_{\gamma_0}}])^{-\langle\gamma_0, \beta\rangle},$$

$$\mathcal{S}_{-}[e^{V_\gamma}] = \mathcal{S}_{+}[e^{V_\gamma}]\, (1+\mathcal{S}_{+}[e^{V_{\gamma_0}}])^{-(\gamma_0,\gamma)},$$

where $\langle\gamma_0, \beta\rangle$ and $(\gamma_0,\gamma)$ are intersection numbers—rendering the Stokes automorphism acting on the Voros symbol algebra as a field automorphism.

Resurgent analysis is realized here: the Borel–Écalle resummed quantization conditions are made ambiguity-free by median resummation, thus restoring global analytic control even for non-Borel summable (Stokes) series.

3. Mutation Theory, Cluster Algebra Structure, and Wall-Crossing

Exact WKB analysis rigidly connects to the combinatorics of cluster algebras. The exchange graph of mutations (e.g., flips and pops of the Stokes triangulations) matches the mutation pattern for seeds in surface-realized cluster algebras (Iwaki et al., 2014). Each Stokes graph mutation corresponds to cluster mutations in the exchange matrix (and hence, variables):

• Flips correspond to changing a diagonal in the triangulation; in the Stokes graph, this is the appearance/disappearance of a saddle trajectory.
• Pops (involving self-folded triangles) are handled with distinguished 'pop' automorphisms.

The periodicity in cluster mutations (e.g., pentagon relations for A2-type clusters) enforces periodicity in the sequence of Stokes automorphisms—yielding quantum dilogarithm-type wall-crossing identities: $$\mathfrak{S}_{\gamma_N} \circ \cdots \circ \mathfrak{S}_{\gamma_1} = \mathrm{id}$$ These structures provide nontrivial constraints on the analytic monodromy data (Voros symbols) and reveal deep interplay among exact WKB, cluster algebras, and wall-crossing phenomena in quantum field theory.

4. Implementation of Exact WKB in Physical Problems

In practice, exact WKB techniques are implemented in a range of contexts, including:

• Quantum spectral problems: Quantization conditions are cast as vanishing of determinants built from Borel-resummed Voros multipliers and monodromy or connection matrix elements.
• Gauge theory and supersymmetric quantum mechanics: Voros symbols correspond to (quantum) periods of Seiberg-Witten curves, with nonperturbative corrections and wall-crossing matching the TBA system and resurgent expansions in the Nekrasov–Shatashvili limit (Imaizumi, 2020).
• Mixed anomalies, spectral degeneracy, and periodicity: In systems on $S^1$ with $N$ minima, the Hilbert space splits according to discrete theta angles, and the quantization condition factorizes over these subspaces; even $N$ cases exhibit ground-state doubling, realizing mixed ’t Hooft anomalies (Sueishi et al., 2021).

Analytic control is achieved by tracking the cycles on the Riemann surface and resumming the formal series for the WKB periods. Ambiguities from divergent expansions are systematically cancelled by nonperturbative contributions, as made fully explicit in the resurgent structure encoded by the DDP and related formulas (Sueishi et al., 2020).

5. Quantization Conditions, Stokes Phenomena, and Borel-Écalle Resummation

The global quantization conditions—generalizing Bohr–Sommerfeld and Gutzwiller's trace formulae—are expressed as algebraic or transcendental equations for energy (or more generally, spectral parameters), written in terms of Borel-resummed Voros symbols. For example, in simple cases: $(1+A)(1+A^{-1}) + AB = 0,$ with $A$ and $B$ representing the exponentials of the action along perturbative (A-cycle) and nonperturbative (B-cycle, e.g. bion) cycles. The full solution involves applying connection matrices at each Stokes curve, keeping precise track of Stokes multipliers and Maslov indices; ambiguities in Borel summation are cancelled by the inclusion of the appropriate nonperturbative saddle contributions.

For systems with quantum deformation or topological sectors, one implements median resummation (e.g., $$S_{\mathrm{med}} = S_+ \mathfrak{S}^{-1/2} = S_- \mathfrak{S}^{1/2}$$) so that the spectrum computed from the Borel-Écalle resummed quantization condition is real and ambiguity-free (Kamata et al., 2021).

6. Extensions: Higher-Order Equations, Discrete Systems, and Quantum Field Theory

Exact WKB analysis extends to difference equations (e.g., $q$-difference, or difference equations arising in block decompositions of path integrals), and higher-order ODEs:

• For $q$-difference equations, such as those governing holomorphic blocks in supersymmetric gauge theories, the q-Borel and q-Laplace transforms, along with associated connection formulas, are precisely orchestrated to achieve analytic continuation and capture Stokes data (Ashok et al., 2019).
• In fourth-order or higher ODEs, such as those appearing in particle production and preheating with CP violation, composite Stokes phenomena—reflecting interference between multiple WKB solutions—are essential in predicting observable asymmetries (Enomoto et al., 2022).

Stokes graphs in these settings often possess complex saddles and branch points whose intersection numbers and mutation patterns reflect in derived wall-crossing and spectral properties.

7. Significance, Conjectures, and Further Connections

Exact WKB analysis provides a framework for analyzing both the classic and non-classic quantization problems. Its transparent treatment of Stokes jumps enables systematic inclusion of all orders in the semiclassical expansion and captures the resurgent bridge between perturbative and nonperturbative sectors. The language of Voros symbols and Stokes automorphisms, connected through cluster algebra combinatorics, brings combinatorial, geometric, and analytic tools into a common platform, enabling new forms of wall-crossing identities, spectral dualities, and anomaly constraints.

Further, the synthesis with the theory of cluster algebras and quantum monodromy is critical to understanding periodicity, spectral network mutations, and the so-called wall-crossing formulae in integrable, supersymmetric, and topological quantum systems. These structures also resonate with recent advances in quantum and enumerative geometry, resurgence theory, and higher Teichmüller theory.

---

In sum, exact WKB analysis is a rigorous, algebraically and geometrically rich framework organizing the global analytic, combinatorial, and resurgent content of quantum differential equations, unifying semiclassical, resurgent, and algebraic perspectives with profound implications for mathematical physics and quantum field theory (Iwaki et al., 2014).