Z4-Extended Thermodynamic Bethe Ansatz

• The Z4-extended TBA equations are a set of nonlinear integral equations that incorporate a four-fold symmetry, extending standard TBA methods for capturing nonperturbative quantum effects.
• They employ sophisticated kernel functions and residue dressings to manage analytic properties and wall-crossing phenomena, ensuring invariant effective central charge.
• This framework bridges quantum integrable models, ODE/IM correspondence, and gauge/Bethe theory, enabling precise study of quantum spectral problems and resurgent analysis.

The ${\mathbb Z}_4$-extended thermodynamic Bethe ansatz (TBA) equations are a class of nonlinear integral equations that generalize the standard TBA formalism by incorporating a discrete ${\mathbb Z}_4$ grading. These equations arise in a variety of settings—including deformed supersymmetric quantum mechanics, quantum integrable systems, spectral problems via the ODE/IM correspondence, and the spectral theory of quantum curves—whenever an underlying four-fold symmetry or structure is present. The ${\mathbb Z}_4$ extension modifies both the system's analytic and combinatoric properties, producing new types of wall-crossing behavior and extended Y- (and T-) systems relevant for both quantum mechanics and two-dimensional quantum field theory.

1. Definition and Structural Features of the $\mathbb{Z}_4$-Extended TBA

The ${\mathbb Z}_4$-extended TBA equations are formulated as an infinite (or highly structured finite) set of coupled nonlinear integral equations for functions $Y_{a,s}(\theta)$, where $a \in \mathbb{Z}_4$ is a label mod 4 and $s$ indexes cycles or sectors. In the context of deformed supersymmetric quantum mechanics, these $Y$-functions encode the Borel-resummed all-orders WKB periods, which characterize the exact, nonperturbative spectrum. The main equations in the minimal chamber (where all classical turning points are real and distinct) take the schematic form

$$\log Y_{a,s}(\theta) = -|m_s|e^{\theta} + m_{a,s}^{(1/2)} + \sum_{s'} \left[ K_{+;s,s'} * L_{a+1,s'} + K_{-;s,s'} * L_{a+3,s'} \right](\theta),$$

where $K_{\pm}$ are kernel functions (e.g., $$K_{+}(\theta) = [1/(4\pi)](1/\cosh\theta + i\tanh\theta)$$), $L_{a,s} = \log(1 + Y_{a,s})$ (sometimes with further dressing or shifts), and the sums run over adjacent cycles or charges. The modulus-4 shift encodes the ${\mathbb Z}_4$ structure. The TBA system is closed by providing asymptotic conditions and, in some cases, algebraic identities among the $Y$-functions.

The extension to ${\mathbb Z}_4$ is necessary to realize and track the full analytic and monodromy discontinuity structure of exact WKB periods, particularly in the presence of odd-power $\hbar$-corrections, phase twists, or as a reflection of the underlying symmetry. The typical $A_n$- or $D_n$-type TBA equations are naturally embedded as specializations or reductions of the extended system.

2. Wall-Crossing and Preservation of $\mathbb{Z}_4$-Structure

Wall-crossing in the TBA context occurs when the analytic structure of the quantum system (such as the number or connectivity of classical turning points) changes as external parameters (e.g., energy, coupling constants) are varied. Across such walls, additional singularities in the convolution kernels can pinch the integration contour, rendering the original TBA system singular or incomplete if unmodified.

In the ${\mathbb Z}_4$-extended TBA, the wall-crossing process is managed by introducing new $Y$-functions corresponding to emerging cycles (for example, those associated with combinations of original cycles, such as $\gamma_{12}$, $\gamma_{32}$, etc.), and by absorbing residue contributions arising due to shifted kernel poles. The dressing of $Y$-functions to absorb such residues typically takes the form

$$Y_{a,1}^{(no\ offset)} = \frac{Y_{a,1}}{1 + Y_{a+3,2}\left(\theta - i\frac{\pi}{2}\right)},$$

and yields a "dressed" TBA system that maintains the ${\mathbb Z}_4$ grading while accounting for all necessary analytic data. Remarkably, the effective central charge, a quantity computed from the integral over pseudo-energies,

$$c_{\rm eff} = \frac{6}{\pi^2} \sum_s \int |m_s| e^\theta \, \tilde{L}_s(\theta) d\theta,$$

remains invariant under wall-crossing, manifesting the integrability and structural robustness of the theory (Ito et al., 8 Oct 2025).

3. Cubic Superpotential and D-type TBA Decomposition

For the cubic superpotential (leading to the quartic double-well potential after deformation), the ${\mathbb Z}_4$-extended TBA equations exhibit notable simplifications due to parity symmetry. In the minimal chamber, three independent cycles (corresponding to the three distinct segments between four real turning points) give rise to three sets of $Y_{a,s}$. Upon imposition of $\mathbb{Z}_2$ symmetry, the TBA equations decouple into two independent $D_3$-type systems. Explicitly, with a suitable identification of $Y$-functions, the equations take the form

$$\log \Psi_{a,1}(\theta) = -|m_1|e^\theta + K_{1,2} * \tilde{L}_{a+1,2}(\theta),$$

$$\log \tilde{Y}_{a+1,2}(\theta) = -|m_2|e^\theta + K_{2,1} * \left[ \tilde{L}_{a,1}(\theta) + \tilde{L}_{a+2,1}(\theta) \right],$$

together with functional equations for the $Y$-system, e.g.,

$$\Psi_{a,1}\left(\theta - \frac{i\pi}{2}\right) \Psi_{a,1}\left(\theta + \frac{i\pi}{2}\right) = 1 + \tilde{Y}_{a+1,2}(\theta).$$

In the maximal chamber, after crossing a wall where turning points pair off into complex conjugates, further cycles are introduced and appropriate normalizations (e.g., $\widehat{Y}_{a,1} = e^{-2\pi i m}Y_{a,1}$) lead back to a pair of $D_3$-type TBA equations, now with different interpretations of the effective cycles. Thus, the $\mathbb{Z}_4$-extension naturally generalizes the $A_3/\mathbb{Z}_2$ TBA equations encountered in the ODE/IM and GMN frameworks (Ito et al., 8 Oct 2025, Ito et al., 8 Jan 2024, Ito et al., 2 Aug 2024).

4. Kernel Structure, Nonperturbative Content, and Analytic Properties

The extended kernels $K_{+}$ and $K_{-}$ in the ${\mathbb Z}_4$-TBA feature both real and imaginary components, like

$$K_{+}(\theta) = \frac{1}{4\pi} \left( \frac{1}{\cosh\theta} + i\tanh\theta \right),$$

which reflect the phase shifts associated with the underlying $\mathbb{Z}_4$ symmetry and the associated rotation of cycles or Stokes sectors in the quantum differential equation. Their analytic properties guarantee that when expanded for large $\theta$, both even and odd powers in $\hbar$ emerge, a signature of the deformed (nontrivial) quantum geometry. The matching of these expansions against Borel-resummed WKB periods (often called Voros periods) ensures that the TBA equations capture both the perturbative and nonperturbative quantum corrections in an exact fashion.

A crucial consistency requirement, especially under wall-crossing, is that the sum of kernel residues and the associated dressing factors do not spoil the analytic continuation properties of the $Y$-system. This ensures that the same set of TBA equations (up to dressing/relabeling) remains valid as parameters are varied, and significant quantities such as the effective central charge are preserved across moduli space (Ito et al., 8 Oct 2025, Ito et al., 8 Jan 2024).

5. Connections to Quantum Integrability and Gauge/Bethe Correspondence

The ${\mathbb Z}_4$-extended TBA equations can be realized as a degenerate limit or symmetry-restricted case of the general fused or "fat hook" TBA equations in AdS/CFT integrability (0902.4458, Cavaglià et al., 2010). The presence of the ${\mathbb Z}_4$ grading is mirrored by the arrangement of $Y_{a,s}$ on a $T$-hook diagram and connects directly with the fusion relations and Hirota dynamics of supersymmetric spin chains and sigma models.

In the context of supersymmetric gauge theories, particularly in the Nekrasov–Shatashvili limit, the ${\mathbb Z}_4$-extension is inherited via the symmetry of the underlying quantum spectral curve, and the TBA equations derived via Mayer cluster expansion and Yang–Yang functional extremization reflect these properties (Meneghelli et al., 2013, Kozlowski et al., 2010). This extension thus governs the quantization conditions of broad classes of algebraically integrable systems, as well as BPS spectra and moduli spaces in higher-dimensional gauge theories (Alexandrov et al., 2010).

6. Implications, Applications, and Further Directions

The ${\mathbb Z}_4$-extended TBA equations provide a uniform nonperturbative framework for several previously distinct problems:

• Exact WKB periods and quantization conditions in polynomial Schrödinger equations with symmetry or $\hbar$-deformed potentials (Ito et al., 8 Oct 2025, Ito et al., 8 Jan 2024).
• Wall-crossing and Stokes automorphism in the resurgent analysis of quantum mechanical and field-theoretic systems.
• The matching of spectral quantities and effective central charges (and thereby with 2d conformal data) across moduli space, even as analytic and geometric structures vary (Ito et al., 2 Aug 2024).
• Natural extension of $A_n$- and $D_n$-type TBA/Y-systems, encoding the presence of fusion, extra symmetry, or flavor in quantum integrable models and their geometric or gauge theory avatars (0902.4458, Cavaglià et al., 2010).
• Realization of ODE/IM correspondences for higher-rank or symmetry-graded quantum systems, facilitating explicit analytical computations in terms of elementary or special functions, and allowing for reductions to simpler systems under parity or other symmetries (Suzuki, 2015).

A plausible implication is that the ${\mathbb Z}_4$-extension represents a universal mechanism for bookkeeping the monodromy (or Stokes data) of the exact spectrum under arbitrary parametric deformations, and may generalize further (to, e.g., ${\mathbb Z}_N$ settings) when studying quantum systems with higher order cyclic symmetries or in the presence of discrete torsion and orbifolding.

7. Formal Summary and Key Equations

Feature	Standard TBA	$\mathbb{Z}_4$-Extended TBA
Y-function structure	$Y_s$	$Y_{a,s}$, $a \in \mathbb{Z}_4$
Kernel type	Real, e.g., $1/\cosh(\theta)$	Complexified, $K_{+}$, $K_{-}$ with phase shifts
Wall-crossing	May introduce new cycles	Additional residue terms, robust $\mathbb{Z}_4$ grading
Central charge under wall-crossing	May jump	Invariant across moduli (with $\mathbb{Z}_4$ TBA prescription)

Prototypical $\mathbb{Z}_4$-extended TBA equation:

$$\log Y_{a,s}(\theta) = -|m_s|e^{\theta} + m_{a,s}^{(1/2)} + K_{+} * L_{a+1,s'} + K_{-} * L_{a+3,s'}$$

with recursive relations (Y-system) inheriting periodicity/branching structure in $a \mapsto a+1$ mod 4.

These mathematical structures underlie integrable structure in a broad range of quantum systems and provide a potent toolset for nonperturbative spectral theory, wall-crossing, and the global paper of moduli spaces in both quantum mechanics and field theory.