Transseries Sectors and Parameters

A transseries enlarges a power series by the exponential, algebraic, and logarithmic sectors required by the governing equation and its global data. Its parameters are not freely adjustable decorations: an ordinary differential equation supplies integration constants, a boundary-value problem fixes them, and a path integral fixes the corresponding cycle coefficients. Stokes jumps change how the same solution is represented, not which physical solution was chosen.

Required background. Borel singularities, lateral sums, and Stokes data supplies the directional sums and discontinuity convention.

Helpful background. Multi-saddle sums and dilute ensembles gives the saddle expansion that often supplies the exponential sectors in semiclassical problems.

The minimal transseries ansatz

For one nonperturbative action $A$ and one parameter $\sigma$, a useful schematic form is

$$\mathcal O(g;\sigma) =\sum_{k=0}^{\infty} \sigma^k e^{-kA/g}g^{k\beta} \sum_{\ell=0}^{L_k}(\log g)^\ell \Phi_{k,\ell}(g), \qquad \Phi_{k,\ell}(g) \sim\sum_{n=0}^{\infty}a_{k,\ell,n}g^n.$$

Each factor has a distinct origin:

• $e^{-kA/g}$ distinguishes exponential scales, often associated with saddle-action differences;
• $g^{k\beta}$ records fluctuation determinants, zero modes, or local Borel exponents;
• $\Phi_{k,\ell}$ is the asymptotic fluctuation series within a sector;
• logarithms can appear through resonance, repeated actions, renormalization, or quasi-zero-mode integrals; and
• $\sigma$ labels a solution or integration cycle in a chosen sectorial basis.

With several independent actions, $kA$ becomes $\boldsymbol k\mathbin{\cdot}\boldsymbol A$ and the parameter becomes $\boldsymbol\sigma^{\boldsymbol k}$. This formal structure is not a promise that every QFT observable has a known finite action set or that all sectors have been identified.

Substitution into the defining equation determines relations among the coefficients. In nonlinear equations, lower sectors source higher ones, so the fluctuation series are not independent. Resonance occurs when different integer combinations of actions coincide; then logarithms or additional parameters can be forced. Costin’s rank-one nonlinear-ODE results state precise hypotheses under which such formal transseries are analyzable and Borel summable in sectors; see Costin 1998, §1 and Theorem 1.

A linear ODE that fixes the entire structure

Consider

$$g^2\frac{\mathrm dy}{\mathrm dg}-y=-g, \qquad g\to0^+.$$

A formal power series $y_0(g)=\sum_{n\ge0}a_n g^n$ gives

$$a_0=0, \qquad a_1=1, \qquad a_n=(n-1)a_{n-1}=(n-1)!,\quad n\ge2.$$

Thus

$$y_0(g)\sim\sum_{n=1}^{\infty}(n-1)!g^n, \qquad \widehat y_0(\zeta) =\sum_{n=1}^{\infty}\frac{\zeta^n}{n} =-\log(1-\zeta).$$

The homogeneous equation has the exact solution $e^{-1/g}$, so the full one-parameter formal solution is

$$y(g;\sigma)=\mathcal S y_0(g)+\sigma e^{-1/g}.$$

No guesswork produced the exponential: it is the homogeneous solution. The logarithmic Borel cut begins at $\zeta=1$. With the chapter’s upper-minus-lower convention,

$$\operatorname{Disc}\widehat y_0(x)=2\pi i, \qquad x>1,$$

and hence

$$\mathcal S_{0^+}y_0- \mathcal S_{0^-}y_0 =\frac{2\pi i}{g}\int_1^\infty e^{-x/g}\,\mathrm dx =2\pi i e^{-1/g}.$$

The same analytic solution is represented on the two sides when

$$\sigma_+=\sigma_- -2\pi i.$$

The equation determines the allowed exponential sector and its normalization; a boundary condition fixes one actual value of the integration constant. For example, specifying $y(g_\ast)=y_\ast$ at a nonsingular $g_\ast$ determines

$$\sigma=e^{1/g_\ast} \left[y_\ast-\mathcal S_\theta y_0(g_\ast)\right]$$

in the chosen sectorial representation. Calling $\sigma$ “arbitrary” after boundary data have been supplied would count the same freedom twice.

From differential equations to path integrals

The ODE analogy transfers only after the global datum is identified. For a finite-dimensional integral or a regulated path integral, the analogue of boundary data is the integration cycle

$$\Gamma=\sum_\alpha n_\alpha\mathcal J_\alpha, \qquad n_\alpha\in\mathbb Z,$$

where $\mathcal J_\alpha$ are downward cycles. The coefficients $n_\alpha$ and the fluctuation normalization determine the sector parameters. Across a Stokes wall the thimble basis and the transseries parameters jump together while $\Gamma$ stays fixed. A formal bridge equation can encode this relation, but it cannot replace the cycle calculation.

In spectral problems, a global quantization condition plays the same role. For degenerate quantum-mechanical minima, uniform WKB relates perturbative and multi-instanton sectors only after the global eigenvalue condition is imposed; Dunne and Ünsal 2014, §§II–III gives an explicit realization. This is why local perturbation theory plus an unspecified $\sigma$ is not yet a spectrum.

The Borel and transseries map distinguishes the analytic jump from the model-specific input that identifies sectors. The exact and rigorous status comparison separates exact ODE statements from extrapolations to QFT.

Normalization and parameter changes

Sector normalizations are conventional. If

$$\Phi_1\longrightarrow c\,\Phi_1, \qquad \sigma\longrightarrow c^{-1}\sigma,$$

the observable is unchanged, while the numerical Stokes constant changes. Meaningful comparisons therefore state:

1. the normalization of $A$ and $g$;
2. the leading coefficient of each $\Phi_k$;
3. the direction and order used in the discontinuity;
4. the boundary condition, quantization condition, or cycle; and
5. the observable to which the transseries belongs.

Different observables of the same theory can require different prefactors or even different visible sectors. A transseries belongs to an equation-and-solution problem, not to a theory name alone.

Boundaries of the ansatz

The displayed ansatz is a useful local form, not complete resurgent algebra. It may need fractional powers, several incommensurate actions, nested exponentials, or infinitely many singular directions. In field theory, renormalization and infinite volume can create additional complications. The later pages test instanton and renormalon interpretations separately rather than treating every $A$ as the action of a real saddle.

Common pitfalls

Parameters are not fitted at every order. Once the equation and global data fix them, changing a parameter to improve a truncated numerical fit changes the solution unless that fitting procedure is itself the boundary condition.

A sector label is not a saddle proof. Exponential scaling can reveal the action scale that a saddle would need, but existence, contour relevance, and fluctuation normalization require separate checks.

Logarithms are not optional clutter. When resonance or a quasi-zero mode forces them, omitting logarithmic sectors makes the substituted equation fail at a definite order.

Exercises

1. Verify directly that $y_0(g)+\sigma e^{-1/g}$ solves the ODE order by order, assuming a lateral Borel sum for $y_0$.

Solution

The recurrence makes $g^2y_0'-y_0=-g$ as a formal identity, and Borel summation preserves the linear differential equation in a common summability sector. For the homogeneous term,

$$g^2\frac{\mathrm d}{\mathrm dg}e^{-1/g} -e^{-1/g}=0.$$

Linearity then proves the statement for every constant $\sigma$.

2. Rescale the one-instanton sector to $\widetilde\Phi_1=c\Phi_1$. Determine the transformed parameter and Stokes constant.

Solution

Keeping $\sigma\Phi_1$ invariant requires $\widetilde\sigma=\sigma/c$. If the old jump is $\Delta\Phi_0=S_{01}e^{-A/g}\Phi_1$, then

$$\Delta\Phi_0 =\frac{S_{01}}c e^{-A/g}\widetilde\Phi_1,$$

so $\widetilde S_{01}=S_{01}/c$. The parameter jump transforms consistently.

References

• Costin, Ovidiu. “On Borel Summation and Stokes Phenomena for Rank-1 Nonlinear Systems of Ordinary Differential Equations.” Duke Mathematical Journal 93 (1998): 289–344. doi:10.1215/S0012-7094-98-09311-5.
• Dunne, Gerald V., and Mithat Ünsal. “Uniform WKB, Multi-Instantons, and Resurgent Trans-Series.” Physical Review D 89 (2014): 105009. arXiv:1401.5202; doi:10.1103/PhysRevD.89.105009.