We present a counting formula that relates the number of physical Bethe states of integrable models with a twisted boundary condition to the number of states in the untwisted or partially twisted limit.

China

Hongfei Shu

Peng Zhao

Rui-Dong Zhu

Hao Zou

Counting Bethe states in twisted spin chains

<p>We study the zero-dimensional prototype of the path integrals in quantum mechanics and quantum field theory, with the action <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mi>σ</mi><mn>2</mn></mfrac><msup><mi>ϕ</mi><mn>2</mn></msup><mo>+</mo><mfrac><mi>λ</mi><mn>4</mn></mfrac><msup><mi>ϕ</mi><mn>4</mn></msup></math>. Using the Lefschetz thimble decomposition and the saddle point expansion, we derive multiple asymptotic formal series of the correlation function associated with the perturbative and nonperturbative saddle points. Furthermore, we reconstruct the exact correlation function employing the Borel resummation. We then consider how to truncate the Dyson-Schwinger equations to compute two-point functions beginning with the perturbation expansion of the correlation functions, analogous to the one obtained from the Feynman diagram in higher dimensions. For the case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>σ</mi><mo>&lt;</mo><mn>0</mn></math>, we find that, although the asymptotic series around the perturbative saddle point is Borel summable, it does not capture the full information. Consequently, contributions from nonperturbative saddle points must be included to ensure a complete truncation procedure.</p>

Feiyu Peng

Truncating Dyson-Schwinger equations based on a Lefschetz thimble decomposition and Borel resummation

We investigate the proposed holographic duality between the TsT transformation of IIB string theory on AdS3 ×  $$\mathcal{N}$$  with NS-NS flux and a single-trace  $$T\overline{T }$$  deformation of the symmetric orbifold CFT. We present a non-perturbative calculation of two-point correlation functions using string theory and demonstrate their consistency with those of the  $$T\overline{T }$$  deformation. The two-point correlation function of the deformed theory on the plane, written in momentum space, is obtained from that of the undeformed theory by replacing h with  $$h+2\frac{\widetilde{\lambda }}{w}p\overline{p }$$ , where h is the spacetime conformal weight,  $$\widetilde{\lambda }$$  is a deformation parameter, p and  $$\overline{p }$$  are the momenta, and w labels the twisted sectors in the deformed symmetric product. At w = 1, the non-perturbative result satisfies the Callan-Symanzik equation for double-trace  $$T\overline{T }$$  deformed CFT derived in [1]. We also perform conformal perturbations on both the worldsheet CFT and the symmetric orbifold CFT as a sanity check. The perturbative and non-perturbative matching between results on the two sides provides further evidence of the conjectured  $${\text{TsT}}/T\overline{T }$$  correspondence.

Wei Cui

Wei Song

Juntao Wang

Correlation functions in the  $${\text{TsT}}/T\overline{T }$$  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.

Japan

Katsushi Ito

TBA equations and exact WKB analysis in deformed supersymmetric quantum mechanics

We study the quasinormal modes (QNM) of the charged C-metric, which physically stands for a charged accelerating black hole, with the help of Nekrasov’s partition function of 4d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 superconformal field theories (SCFTs). The QNM in the charged C-metric are classified into three types: the photon-surface modes, the accelerating modes and the near-extremal modes, and it is curious how the single quantization condition proposed in [1] can reproduce all the different families. We show that the connection formula encoded in terms of Nekrasov’s partition function captures all these families of QNM numerically and recovers the asymptotic behavior of the accelerating and the near-extremal modes analytically. Using the connection formulae of different 4d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 SCFTs, one can solve both the radial and the angular parts of the scalar perturbation equation respectively. The same algorithm can be applied to the de Sitter (dS) black holes to calculate both the dS modes and the photon-sphere modes.

Yang Lei

Kilar Zhang

Quasinormal modes of C-metric from SCFTs

We first derive the boundary theory from the U(1) Chern-Simons theory. The boundary action on an n-sheet manifold appears from its back-reaction of the Wilson line. The reason is that the U(1) Chern-Simons theory can provide an exact effective action when introducing the Wilson line. The Wilson line in the pure AdS$_{3}$ Einstein gravity is equivalent to entanglement entropy in the boundary theory up to classical gravity. The U(1) Chern-Simons theory deviates by a self-interaction term from the gauge formulation on the boundary. We also compare the Hayward term in the SL(2) Chern-Simons formulation to the Wilson line approach. Introducing two wedges can reproduce the entanglement entropy for a single interval at the classical level. We propose quantum generalization by combining the bulk and Hayward terms. The quantum correction of the partition function vanishes. In the end, we calculate the entanglement entropy for a single interval. The pure AdS$_{3}$ Einstein gravity theory shows a shift of central charge by 26 at the one-loop level. The U(1) Chern-Simons theory does not show a shift from the quantum effect. The result is the same in the weak gravitational constant limit. The non-vanishing quantum correction shows that the Hayward term is incorrect.

Xing Huang

South Africa

South Korea

Chen-Te Ma

Sweden

United States

Chih-Hung Wu

U(1) CS theory vs SL(2) CS formulation: Boundary theory and Wilson line

We study scattering amplitudes in the shadow conformal primary basis, which satisfies the same defining properties as the original conformal primary basis and has many advantages over it. The shadow celestial amplitudes exhibit locality manifestly on the celestial sphere, and behave like correlation functions in conformal field theory under the operator product expansion (OPE) limit. We study the OPE limits for three-point shadow celestial amplitude, and general 2 → n − 2 shadow celestial amplitudes from a certain class of Feynman diagrams. In particular, we compute the conformal block expansion of the s-channel four-point shadow celestial amplitude of massless scalars at tree-level, and show that the expansion coefficients factorize as products of OPE coefficients.

Chi-Ming Chang

Wen-Jie Ma

Shadow celestial amplitudes

We compute the minimal area of a string worldsheet ending on two infinite periodic light-like Wilson lines in the AdS3 boundary, which is dual to the first non-planar correction to the gluon scattering amplitude in   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 SYM at the strong coupling. Using the connection between the Hitchin system and the thermodynamic Bethe ansatz (TBA) equations, we present an analytic method to compute the minimal area surface and express the non-trivial part of the minimal area in terms of the free energy of the TBA-like equations. Given the cross ratios as inputs, the area computed from the TBA-like equations matches that calculated using the numerical integration.

Hao Ouyang

TBA-like equations for non-planar scattering amplitude/Wilson lines duality at strong coupling

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.