TBA and wall-crossing for deformed supersymmetric quantum mechanics

TL;DR

The paper shows that deformed supersymmetric quantum mechanics with an $areta$-corrected polynomial potential has exact WKB periods described by a ${ t Z}_4$-extended TBA, which persists across wall-crossing in moduli space. Focusing on the cubic superpotential, the authors reveal two decoupled ${ m D}_3$-type TBA systems in both the minimal and maximal chambers, linking the spectral problem to GMN TBA equations for $(A_1,D_3)$-type AD theories. At the monomial point, the system again reduces to two ${ m D}_3$-type TBA, providing a ${A_3}/{ t Z}_2$-extension framework. The effective central charge remains invariant under wall-crossing, underscoring a robust nonperturbative structure with connections to 2d CFT/D-brane interpretations and AD theory, and suggesting avenues for analytic continuation and excited-state generalizations.

Abstract

We study the deformed supersymmetric quantum mechanics with a polynomial superpotential with $\hbar$-correction. In the minimal chamber, where all turning points are real and distinct, it was shown that the exact WKB periods obey the ${\mathbb Z}_4$-extended thermodynamic Bethe ansatz (TBA) equations of the undeformed potential. By changing the energy parameter above/below the critical points, the turning points become complex, and the moduli are outside of the minimal chamber. We study the wall-crossing of the ${\mathbb Z}_4$-extended TBA equations by this change of moduli and show that the ${\mathbb Z}_4$-structure is preserved after the wall-crossing. In particular, the TBA equations for the cubic superpotential are studied in detail, where there are two chambers (minimal and maximal). At the maximally symmetric point in the maximal chamber, the TBA system becomes the two sets of the $D_3$-type TBA equations, which are regarded as the ${\mathbb Z}_4$-extension of the $A_3/{\mathbb Z}_2$-TBA equation.

Figures (5)

• Figure 1: In Fig.(a), the wall on the $E$ moduli space for a fixed value of $u_1= 1/4$ is shown. The region inside the red curve denotes the minimal chamber, while the region outside corresponds to the maximal chamber. The wall on the $u_1$ moduli space for the fixed value $E=31/32$ is shown in Fig.(b). Here the minimal chamber extends from the red line to infinity, while the maximal chamber is the bounded region near the origin.
• Figure 2: A path from the minimal to the maximal chamber by deforming the turning points of $Q_0(x)$ (red dots). The wavy lines denote the branch cut on the WKB curve. In each chamber, the one-cycles correspond to the Y-functions. In each process of wall-crossing, a new one-cycle (red cycle) and a new Y-function appear.
• Figure 3: $m=0.47$
• Figure 4: $m=0.48$
• Figure 5: $m=0.49$