---
title: Voros Coefficient in Exact WKB Analysis
url: https://www.emergentmind.com/topics/voros-coefficient
type: topic
---

# Voros Coefficient in Exact WKB Analysis

A Voros coefficient is a regularized WKB integral attached to a path or cycle on a spectral curve. In exact WKB analysis it is typically defined by subtracting the divergent leading contribution from a Riccati or WKB one-form and integrating the remainder between a turning point and a singular endpoint, or around a closed cycle; it measures the discrepancy between natural normalizations of formal solutions and controls Stokes or parametric Stokes phenomena [1303.3603], [2108.06995]. In quantum-curve formulations its exponentials are the Voros symbols, and in several hypergeometric, Painlevé, and Heun-type problems these quantities admit explicit Bernoulli- or Gamma-function descriptions [1805.10945], [2104.13751], [2605.06079].

## 1. Definition in exact WKB theory

For the third Painlevé equation of type \(D_6\), a Voros coefficient is defined as a regularized integral of the odd WKB term with the divergent leading part subtracted. For a path \(\Gamma(\tau,\infty)\) from a turning point or simple pole \(\tau\) to \(\infty\),
\[
W_{\infty}({\bf c},\eta)=\int_{\Gamma(\tau,\infty)}\bigl(R_{\rm odd}(t,{\bf c},\eta)-\eta R_{-1}(t,{\bf c})\bigr)\,dt,
\]
and similarly for paths ending at the double-pole branches \(0_{c_\ast}\) [1303.3603]. Here \(R_{\rm odd}\) is the odd part of the formal Riccati solution, and the subtraction of \(\eta R_{-1}\) is the regularization that makes the integral convergent at the singular endpoint [1303.3603].

In the quantum-curve and topological-recursion setting, the same object is formulated as an integral of the odd TR/WKB one-form. For cycles \(\gamma\in H_1(\Sigma,\mathbb Z)\) and relative paths \(\beta\in H_1(\Sigma,D_\infty,\mathbb Z)\),
\[
V_\gamma(\hbar):=\int_\gamma dS^{\rm odd}_{\rm TR}(z,\hbar),\qquad
V_\beta(\hbar):=\int_\beta dS^{\rm odd}_{\rm TR,\ge 1}(z,\hbar),
\]
where \(dS^{\rm odd}_{\rm TR,\ge 1}\) is obtained from \(dS^{\rm odd}_{\rm TR}\) by subtracting the \(\hbar^{-1}\) and \(\hbar^0\) endpoint singularities [2108.06995]. This formulation makes explicit the distinction between cycle Voros coefficients and path Voros coefficients.

A closely related convention appears for second-order and higher-order hypergeometric equations, where the coefficient is written using the full Riccati solution \(S=\sum_{m\ge -1}\hbar^mS_m\) and regularized by subtracting the non-integrable terms \(\hbar^{-1}S_{-1}+S_0\):
\[
V_\gamma(\hbar)=\int_\gamma \left(S(x,\hbar)-\hbar^{-1}S_{-1}(x)-S_0(x)\right)\,dx
=\sum_{m=1}^\infty \hbar^m\int_\gamma S_m(x)\,dx.
\]
This is the convention used for the Weber equation and for the confluent family of Gauss hypergeometric equations [1805.10945], [1810.02946].

The Airy equation provides the local model in which the coefficient is effectively trivial after canonical normalization. In that case the exact connection formula is
\[
\Psi_{+}^{\rm I}=\Psi_{+}^{\rm II}+i\,\Psi_{-}^{\rm II},\qquad
\Psi_{-}^{\rm I}=\Psi_{-}^{\rm II},
\]
with no extra factor of the form \(e^{V(\eta)}\); the connection data are exhausted by the universal Stokes constant \(i\) [2205.02988]. This suggests that nontrivial Voros coefficients arise from genuinely global normalization data rather than from the universal local simple-turning-point model.

## 2. Normalization, transseries, and Stokes phenomena

The fundamental role of a Voros coefficient is to compare different normalizations of the same formal or resummed solution. For \((P_{\rm III'})_{D_6}\), if \(\lambda_\tau\) denotes the transseries normalized at a turning point \(\tau\) and \(\lambda_\infty\) the one normalized at \(\infty\), then
\[
\tilde{\lambda}^{(1)}_\tau = e^{W_\infty({\bf c},\eta)}\tilde{\lambda}^{(1)}_\infty,
\qquad
\lambda_\tau(t,{\bf c},\eta;\alpha)=
\lambda_\infty\bigl(t,{\bf c},\eta;\alpha e^{W_\infty({\bf c},\eta)}\bigr).
\]
Thus the Voros coefficient is the multiplicative renormalization of the instanton parameter between two normalizations [1303.3603].

This normalization role becomes decisive when Stokes geometry degenerates. For \((P_{\rm III'})_{D_6}\), triangle-type and loop-type degenerations occur when certain parameter combinations become purely imaginary, and the corresponding Bernoulli-series expressions for the Voros coefficients cease to be Borel summable [1303.3603]. The Borel sums of the basic building blocks satisfy jump formulas
\[
{\mathcal S}_{+}[e^{\cal F}(c,\eta)] = (1+e^{2\pi i c\eta})\,{\mathcal S}_{-}[e^{\cal F}(c,\eta)],
\qquad
{\mathcal S}_{+}[e^{\cal G}(c,\eta)] = (1-e^{2\pi i c\eta})\,{\mathcal S}_{-}[e^{\cal G}(c,\eta)],
\]
and these factors become the connection coefficients for the transseries parameter [1303.3603].

In exact WKB on marked bordered surfaces, the exponentials of Voros coefficients are the Voros symbols. For a cycle \(\gamma\) and a path \(\beta\),
\[
V_\gamma=\oint_\gamma S_{\mathrm{odd}}(z,\hbar)\,dz,\qquad
W_\beta=\int_\beta S_{\mathrm{odd}}^{\mathrm{reg}}(z,\hbar)\,dz,
\]
and the corresponding symbols are \(e^{V_\gamma}\) and \(e^{W_\beta}\) [1802.05479]. The Borel sums of the cycle symbols associated with arcs of the WKB triangulation are identified with Fock–Goncharov coordinates of framed \(PGL_2(\mathbb C)\)-local systems, and these Borel sums admit multivalued meromorphic continuation to all of \(\mathbb C^*\), branched only at the origin [1802.05479]. This places the Voros coefficient within the monodromy and cluster-geometric structure of exact WKB.

A more recent application appears in exact-WKB quantization on time-dependent backgrounds. There the Voros coefficient is
\[
\mathcal{V}_{\rm voros}\equiv \frac{1}{2}\int_{\gamma_{\sigma_0,\tau_0}}
\left(S_{\rm odd}-\eta S_{-1}\right)\,dx,
\]
and it relates turning-point-normalized exact WKB solutions to asymptotic-point-normalized ones by
\[
\psi_{\pm,\tau_0}=\exp(\pm\mathcal{V}_{\rm voros})\,\psi^{(\sigma_0)}_\pm.
\]
The paper states that without this factor the exact WKB solutions generally do not match the asymptotic vacuum normalization and the resulting mode functions can fail quantization [2509.19194]. This suggests that the Voros coefficient is not merely a higher-order correction but a normalization datum required for global physical consistency.

## 3. Explicit evaluation and Bernoulli structures

A major feature of Voros coefficients is that, in many integrable examples, they collapse to universal Bernoulli series. For \((P_{\rm III'})_{D_6}\), all Voros coefficients are expressed in terms of
\[
{\cal F}(c,\eta)=\sum_{n=1}^{\infty}\frac{2^{1-2n}-1}{2n(2n-1)}\,B_{2n}\,(c\eta)^{1-2n},
\qquad
{\cal G}(c,\eta)=\sum_{n=1}^{\infty}\frac{B_{2n}}{2n(2n-1)}\,(c\eta)^{1-2n},
\]
with \(c_p=(c_\infty+c_0)/2\) and \(c_m=(c_\infty-c_0)/2\) [1303.3603]. For example,
\[
W_{\infty_{1,\pm}}({\bf c},\eta)=W_{\infty_{2,\pm}}({\bf c},\eta)=\pm{\cal F}(c_p,\eta),
\qquad
W_{\infty_{3,\pm}}({\bf c},\eta)=W_{\infty_{4,\pm}}({\bf c},\eta)=\pm{\cal F}(c_m,\eta),
\]
while the double-pole coefficients involve linear combinations of \({\cal F}\) and \({\cal G}\) [1303.3603]. These formulas are derived from difference equations induced by Bäcklund transformations rather than by direct integration.

For the Weber equation, the Voros coefficient has an explicit Bernoulli-polynomial expansion,
\[
V(\lambda,\nu;\hbar)=\sum_{m=1}^{\infty}
\frac{B_{m+1}\big((\nu+1)/2\big)}{m(m+1)}
\left(\frac{\hbar}{\lambda}\right)^m,
\]
and its regularized version is a finite difference of the topological-recursion free energy:
\[
V_{\rm reg}(\lambda,\nu;\hbar)=
F\!\left(\hat{\lambda}+\frac{\hbar}{2};\hbar\right)-
F\!\left(\hat{\lambda}-\frac{\hbar}{2};\hbar\right),
\qquad
\hat{\lambda}=\lambda-\frac{\hbar\nu}{2}.
\]
The same paper gives
\[
F_g(\lambda)=\frac{B_{2g}}{2g(2g-2)}\frac{1}{\lambda^{2g-2}}
\qquad (g\ge 2),
\]
showing that the coefficient is controlled by the same Bernoulli structure as the free energy [1805.10945].

The second part of that program extends the same pattern to the confluent family of Gauss hypergeometric equations. For each singular point \(j\),
\[
V^{(j)}_{\mathrm{reg}}(\lambda,\nu;\hbar)=
F\!\left(\lambda+\frac{\hbar}{2}\delta_j;\hbar\right)-
F\!\left(\lambda-\frac{\hbar}{2}\delta_j;\hbar\right),
\]
and the coefficients \(V_m^{(j)}\) are given explicitly by Bernoulli polynomials evaluated at half-shifted combinations of the \(\nu\)-parameters [1810.02946]. The paper emphasizes that different Voros coefficients of the same differential equation arise from different half-\(\hbar\) parameter shifts of a single free energy [1810.02946].

For the generalized hypergeometric equation \({}_N F_{N-1}\) with a large parameter, Voros coefficients are defined separately at \(x=0\) and \(x=\infty\), for a pair of characteristic sheets \((j,k)\):
\[
V^{(j,k)}_\varrho=
\int_\varrho^\tau
\left(S^{(\varrho,j,k)}_{\rm odd}-S^{(\varrho,j,k)}_{{\rm odd},\le 0}\right)\,dx.
\]
The explicit formulas are finite sums of Bernoulli-polynomial terms, and their Borel summability is determined by the sign of the real parts of the relevant parameter differences [2104.13751]. The corresponding Borel sums are given by Gamma-function expressions, which is a standard exact-WKB signature of regularized normalization factors [2104.13751].

## 4. Quantum curves, topological recursion, and BPS structures

In the topological-recursion framework, Voros coefficients become structural rather than merely auxiliary. For hypergeometric-type spectral curves, cycle coefficients satisfy the exact formula
\[
V_\gamma(\hbar)=\frac{Z(\gamma)}{\hbar}-\pi i\,\nu(\gamma),
\]
while path coefficients admit the BPS-sum expansion
\[
V_{\beta,k}=
\sum_{\gamma\in\Gamma\atop Z(\gamma)\in H}
\frac{B_{k+1}(\gamma)}{k(k+1)}\,\Omega(\gamma)\,(\beta,\gamma)
\left(\frac{2\pi i}{Z(\gamma)}\right)^k.
\]
Here the Bernoulli-polynomial weights \(B_{k+1}(\gamma)\) depend on \(\nu(\gamma)\), and the sum is organized by the BPS spectrum \(\Omega(\gamma)\) [2108.06995].

The exponentials of the Borel-resummed Voros coefficients satisfy the same jump formula as the BPS automorphism. The paper proves that the Borel sums of cycle and path Voros symbols solve the almost-doubled BPS Riemann–Hilbert problem, with solution
\[
X^{\rm Vor}_{\ell,\gamma}(\hbar)=\sigma(\gamma)\,\mathcal S_\ell e^{V_\gamma(\hbar)},
\qquad
X^{\rm Vor}_{\ell,\beta}(\hbar)=\sigma(\beta)\,\mathcal S_\ell e^{V_\beta(\hbar)}.
\]
This identifies the Voros symbols with the meromorphic functions required by Bridgeland’s formalism [2108.06995].

A further structural statement is that the path coefficients define a closed one-form on parameter space. If \(\{\beta_s\}_{s\in P_{\rm ev}}\) is the natural basis, then
\[
\omega_k:=\sum_{s\in P_{\rm ev}}2\pi i\,V_{\beta_s,k}\,dm_s,
\qquad
\omega^\bullet:=\sum_{k\ge1}\hbar^k\omega_k,
\]
and there exists a Voros potential \(\phi=\sum_{k\ge1}\hbar^k\phi_k\) such that
\[
\omega^\bullet=2\pi i\, d_{M^\bullet}\phi.
\]
The BPS \(\tau\)-function is then
\[
\tau_{\mathrm{BPS},\ell}=c_\ell\,\mathcal S_\ell e^{-\partial_\hbar\phi},
\]
and at a special quantization parameter this agrees, up to a simple factor, with the Borel sum of the topological recursion partition function \(Z_{\rm TR}\) [2108.06995]. This suggests a broad reinterpretation of Voros coefficients as differential-geometric and wall-crossing data on parameter space.

## 5. Higher-order equations, Heun-type systems, and physical applications

Voros coefficients persist beyond second-order Schrödinger form. For third-order scalar equations associated with degenerations of the 2-dimensional Garnier system, the coefficient is defined by the regularized integral
\[
V_{\gamma_{b_1,b_2}}(\hbar)=
\int_{\gamma_{b_1,b_2}}
\left(S(x,\hbar)-\hbar^{-1}S_{-1}(x)-S_0(x)\right)\,dx.
\]
For the \((1,4)\) quantum curve, the paper proves
\[
V_{\mathrm{reg}}(\lambda_\infty,t,\nu_\infty;\hbar)=
F(\hat\lambda_\infty+\hbar,t;\hbar)-F(\hat\lambda_\infty,t;\hbar),
\]
and derives the explicit Bernoulli-polynomial expansion
\[
V(\lambda_\infty,t,\nu_\infty;\hbar)=
\sum_{m=1}^{\infty}\frac{B_{m+1}(\nu_\infty)}{m(m+1)}
\left(\frac{\hbar}{\lambda_\infty}\right)^m.
\]
For the \((2,3)\) curve, by contrast, the Voros coefficient vanishes identically [2005.08957]. The coexistence of a nontrivial and a zero example within the same framework shows that nontriviality is a global property of the quantum curve rather than a formal inevitability.

For the Heun equation and all of its confluent equations, the relevant exact-WKB quantity is formulated as a Voros period rather than a Voros coefficient:
\[
V_\gamma=\sum_{m\ge -1}\hbar^m\oint_\gamma S_m(x)\,dx.
\]
The paper imposes
\[
\exp(V_\gamma)=\pm\exp\!\left(\frac{2\pi i\nu}{\hbar}\right),
\]
or equivalently
\[
V_\gamma=\frac{2\pi i\nu}{\hbar}-d\pi i,
\]
and uses this condition to determine formal series expansions of the accessory parameter for Heun and every confluent Heun equation in its table [2605.06079]. The spectral curves in these cases have genus \(1\), and the paper gives a detailed prescription for choosing the vanishing cycle that matches the classical regular or irregular conformal block through the accessory parameter [2605.06079]. This suggests that, for genus-one problems, the global period is the appropriate analogue of the more local Voros coefficient of genus-zero hypergeometric systems.

The 2025 paper on time-dependent backgrounds supplies a distinct application. There the Voros coefficient renormalizes exact WKB solutions from turning-point normalization to asymptotic singular-point normalization,
\[
\psi_{\pm,\tau_i}=
\exp\!\left(\pm \mathcal{V}_{\rm voros}^{(\sigma_i)}\right)\psi_\pm^{(\sigma_i)},
\]
and enters the evolution matrix for mode functions [2509.19194]. The paper states that if one were to use the turning-point-normalized solutions directly, the amplitudes would differ by the values of the Voros coefficients compared to the WKB approximation and quantization would fail [2509.19194]. In this application the coefficient is part of the exact normalization of the quantum state rather than only a monodromy invariant.

## 6. Terminology and common confusions

The term “Voros” is used in several mathematically unrelated senses. The following distinctions are explicit in the cited literature.

| Usage of “Voros” | Object studied | Relation to Voros coefficient |
|---|---|---|
| Exact WKB / quantum curves | Regularized WKB integral or its exponential | The standard setting |
| Noncommutative geometry | Voros star-product | Terminological mismatch |
| Analytic number theory | Voros criterion for RH | Different object |

The paper “Noncommutative inspired Schwarzschild black hole, Voros product and Komar energy” is not about a Voros coefficient in the exact-WKB sense. It studies the Voros star-product and states explicitly that there is no introduction, definition, computation, or use of any “Voros coefficient” in the paper [1212.4049]. The same terminological mismatch holds for “Spinors and Voros star-product for Group Field Theory: First Contact,” which studies the Voros star-product on the noncommutative \(\mathbb R^3\) dual to \(SU(2)\), not an exact-WKB coefficient [1107.5693].

A different mismatch occurs in the number-theoretic paper “Analysis of Voros criterion,” which studies a Voros-type criterion for the Riemann hypothesis in terms of zero sums and derivatives of \(\ln \xi(z)\) at \(z=\tfrac12\), and explicitly does not introduce a standard standalone object named “the Voros coefficient” in the WKB or spectral sense [1407.5758].

Even within exact WKB, terminology varies. Some papers reserve “Voros coefficient” for open-path normalization integrals and use “Voros period” for closed-cycle integrals, especially on higher-genus spectral curves [2605.06079]. Others package both path and cycle cases under the common language of Voros coefficients and distinguish them by the underlying homology class [2108.06995]. The most stable invariant across these usages is not the name but the function: a Voros coefficient or Voros period is a regularized WKB integral encoding normalization, monodromy, and Stokes data.

Source: https://www.emergentmind.com/topics/voros-coefficient