<p>We study the deformed supersymmetric quantum mechanics with a polynomial superpotential with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ℏ</mi></math> correction. In the minimal chamber, where all turning points are real and distinct, it was shown that the exact Wentzel–Kramers–Brillouin periods obey the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="double-struck">Z</mi><mn>4</mn></msub></math>-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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="double-struck">Z</mi><mn>4</mn></msub></math>-extended TBA equations by this change of moduli and show that the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="double-struck">Z</mi><mn>4</mn></msub></math> 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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>D</mi><mn>3</mn></msub></math>-type TBA equations, which are regarded as the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="double-struck">Z</mi><mn>4</mn></msub></math> extension of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>A</mi><mn>3</mn></msub><mo>/</mo><msub><mi mathvariant="double-struck">Z</mi><mn>2</mn></msub></math>-type TBA equation.</p>

China

Hongfei Shu

Japan

Katsushi Ito

Jingjing Yang

Thermodynamic Bethe ansatz and wall crossing for deformed supersymmetric quantum mechanics

We construct TBA equations for D-type Argyres-Douglas theories with an SU(2) flavor symmetry based on their spectral networks. We show that the solutions of these TBA equations agree with the quantum periods of the corresponding quantum Seiberg-Witten curves defined in the Nekrasov-Shatashvili limit of the Omega background, including a centrifugal correction. We study the variety of TBA systems across the Coulomb branch moduli space and find that they correspond to the Dynkin diagrams of $\textit{D}$ Lie algebras in the minimal chamber, and reproduce the TBA equations for reflectionless D scattering theories at the maximally symmetric point. Numerical computations demonstrate that the quantum periods obtained from the Borel-Padé resummation and their WKB expansions are in agreement with the solutions of the TBA equations.

TBA equations and quantum periods for D-type Argyres-Douglas theories

We study the ODE/IM correspondence between two-dimensional <math><mi>W</mi><msub><mrow><mi>A</mi></mrow><mrow><mi>r</mi></mrow></msub></math>/<math><mi>W</mi><msub><mrow><mi>D</mi></mrow><mrow><mi>r</mi></mrow></msub></math>-type conformal field theories and the higher-order ordinary differential equations (ODEs) obtained from the affine Toda field theories associated with <math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>r</mi></mrow><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msubsup><mo>/</mo><msubsup><mrow><mi>D</mi></mrow><mrow><mi>r</mi></mrow><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msubsup></math>-type affine Lie algebras. We calculate the period integrals of the WKB solution to the ODE along the Pochhammer contour, where the WKB expansions correspond to the classical conserved currents of the Drinfeld-Sokolov integrable hierarchies. We also compute the integrals of motion for <math><mi>W</mi><msub><mrow><mi>A</mi></mrow><mrow><mi>r</mi></mrow></msub></math>(<math><mi>W</mi><msub><mrow><mi>D</mi></mrow><mrow><mi>r</mi></mrow></msub></math>) algebras on a cylinder. Their eigenvalues on the vacuum state are confirmed to agree with the period integrals up to the sixth order. These results generalize the ODE/IM correspondence to higher-order ODEs and can be used to predict higher-order integrals of motion.

Mingshuo Zhu

Integrals of motion in conformal field theory with -symmetry and the ODE/IM correspondence

We study the spectral problem in deformed supersymmetric quantum mechanics with polynomial superpotential by using the exact WKB method and the TBA equations. We apply the ODE/IM correspondence to the Schrödinger equation with an effective potential deformed by integrating out the fermions, which admits a continuous deformation parameter. We find that the TBA equations are described by the ℤ4-extended ones. For cubic superpotential corresponding to the symmetric double-well potential, the TBA system splits into the two D 3-type TBA equations. We investigate in detail this example based on the TBA equations and their analytic continuation as well as the massless limit. We find that the energy spectrum obtained from the exact quantization condition is in good agreement with the diagonalization approach of the Hamiltonian.

TBA equations and exact WKB analysis in deformed supersymmetric quantum mechanics

Abstract We apply the exact Wentzel–Kramers–Brillouin (WKB) analysis to a couple of 1D Schrödinger-type equations reduced from the Stark effect of hydrogen in a uniform electric field. By introducing Langer’s modification and incorporating the Stokes graphs, we prove the exactness of the Bohr–Sommerfeld quantization conditions for the Borel-resummed quantum WKB periods in the specific parameter regions of the electric field intensity and magnetic quantum number. It is also found these quantization conditions get modified with an additional suppressed contribution when the parameters vary beyond the specific regions. We also present thermodynamic Bethe ansatz (TBA) equations governing the quantum periods in the absence of Langer’s modification and discuss its wall-crossing and analytic continuation. Numerical calculations are conducted to compare the complex resonant frequencies from our quantization conditions against ones from the Riccati–Padé method; the TBA equations are also confirmed by comparing its expansions with all-order quantum periods.

Exact WKB Analysis and TBA Equations for the Stark Effect Quartic oscillator

We study the WKB analysis of the solutions to the linear problem for a modified affine Toda field equation, which is equivalent to the higher-order ordinary differential equation (ODE) studied in the ODE/IM correspondence. After gauge transformation, we diagonalize the flat connection of the linear problem to reduce the latter to a set of independent first-order linear differential equations. We explicitly perform this procedure for classical affine Lie algebras with lower ranks. In particular, we study the WKB solutions of the   <math> <msubsup> <mi>D</mi> <mi>r</mi> <mfenced> <mn>1</mn> </mfenced> </msubsup> </math>  $$ {D}_r^{(1)} $$ - and   <math> <msubsup> <mi>D</mi> <mrow> <mi>r</mi> <mo>+</mo> <mn>1</mn> </mrow> <mfenced> <mn>2</mn> </mfenced> </msubsup> </math>  $$ {D}_{r+1}^{(2)} $$ -type linear problems, which correspond to the higher-order ODEs with the pseudo-differential operator. The diagonalized connection is obtained from the Riccati equations of the adjoint linear problem and related to the conserved currents of the integrable hierarchy constructed by Drinfeld and Sokolov up to total derivatives.

WKB analysis of the linear problem for modified affine Toda field equations

We study the linear differential system associated with the supersymmetric affine Toda field equations for affine Lie superalgebras, which has a purely odd simple root system. For an affine Lie algebra, the linear problem modified by conformal transformation leads to an ordinary differential equation (ODE) that provides the functional relations in the integrable models. This is known as the ODE/IM correspondence. For the affine Lie superalgebras, the linear equations modified by a superconformal transformation are shown to reduce to a couple of ODEs for each bosonic subalgebra. In particular, for <math><mi>o</mi><mi>s</mi><mi>p</mi><msup><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></msup></math>, the corresponding ODE becomes the second-order ODE with squared potential, which is related to the <math><mi>N</mi><mo>=</mo><mn>1</mn></math> supersymmetric minimal model via the ODE/IM correspondence. We also find ODEs for classical affine Lie superalgebras with purely odd simple root systems.

ODE/IM correspondence and supersymmetric affine Toda field equations

We study the WKB periods for the third order ordinary differential equation (ODE) with polynomial potential, which is obtained by the Nekrasov-Shatashvili limit of (<math><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>N</mi></mrow></msub></math>) Argyres-Douglas theory in the Omega background. In the minimal chamber of the moduli space, we derive the Y-system and the thermodynamic Bethe ansatz (TBA) equations by using the ODE/IM correspondence. The exact WKB periods are identified with the Y-functions. Varying the moduli parameters of the potential, the wall-crossing of the TBA equations occurs. We study the process of the wall-crossing from the minimal chamber to the maximal chamber for <math><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math> and <math><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>)</mo></math>. When the potential is a monomial type, we show the TBA equations obtained from the (<math><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub></math>) and (<math><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>3</mn></mrow></msub></math>)-type ODE lead to the <math><msub><mrow><mi>D</mi></mrow><mrow><mn>4</mn></mrow></msub></math> and <math><msub><mrow><mi>E</mi></mrow><mrow><mn>6</mn></mrow></msub></math>-type TBA equations respectively.

Takayasu Kondo

Wall-crossing of TBA equations and WKB periods for the third order ODE

We study the WKB periods for the (r + 1)-th order ordinary differential equation (ODE) which is obtained by the conformal limit of the linear problem associated with the   <math> <msubsup> <mi>A</mi> <mi>r</mi> <mfenced> <mn>1</mn> </mfenced> </msubsup> </math>  $$ {A}_r^{(1)} $$  affine Toda field equation. We compute the quantum corrections by using the Picard-Fuchs operators. The ODE/IM correspondence provides a relation between the Wronskians of the solutions and the Y-functions which satisfy the thermodynamic Bethe ansatz (TBA) equation related to the Lie algebra A  r . For the quadratic potential, we propose a formula to show the equivalence between the logarithm of the Y-function and the WKB period, which is confirmed by solving the TBA equation numerically.

Kohei Kuroda

Sweden

WKB periods for higher order ODE and TBA equations

We study the quantum Seiberg-Witten periods of <math><mi>N</mi><mo>=</mo><mn>2</mn></math> superconformal field theories which are obtained by taking the scaling limit of <math><mi>N</mi><mo>=</mo><mn>2</mn></math> <math><mi>S</mi><mi>U</mi><mo>(</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>)</mo></math> SQCD around the superconformal fixed point. The quantum Seiberg-Witten curves of these superconformal field theories are shown to be classified into the Schrödinger type and the SQCD type, which depend on flavor symmetry at the fixed point. We study the quantum periods and compute the differential operators which relate the quantum periods to the classical ones up to the fourth-order in the deformation parameter.