65 research outputs found

    ODE/IM correspondence and the Argyres-Douglas theory

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    We study the quantum spectral curve of the Argyres-Douglas theories in the Nekrasov-Sahashvili limit of the Omega-background. Using the ODE/IM correspondence we investigate the quantum integrable model corresponding to the quantum spectral curve. We show that the models for the A2NA_{2N}-type theories are non-unitary coset models (A1)1×(A1)L/(A1)L+1(A_1)_1\times (A_1)_{L}/(A_1)_{L+1} at the fractional level L=22N+12L=\frac{2}{2N+1}-2, which appear in the study of the 4d/2d correspondence of N=2{\cal N}=2 superconformal field theories. Based on the WKB analysis, we clarify the relation between the Y-functions and the quantum periods and study the exact Bohr-Sommerfeld quantization condition for the quantum periods. We also discuss the quantum spectral curves for the D and E type theories.Comment: 28 pages, 1 figure. Typos corrected, a reference is added. Published versio

    T-duality to Scattering Amplitude and Wilson Loop in Non-commutative Super Yang-Mills Theory

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    We first perform bosonic T-duality transformation on one of the marginal TsT (T-duality, shift, T-duality)-deformed AdS5×S5AdS_5\times S_5 spacetime, which corresponds to 4D N=4\mathcal{N}=4 non-commutative super Yang-Mills theory (NCSYM). We then construct the solution to killing spinor equations of the resulting background, and perform the fermionic T-duality transformation. The final dual geometry becomes the usual AdS5×S5AdS_5\times S_5 but with the constant NS-NS B-field depending on the non-commutative parameter. As applications, we study the gluon scattering amplitude and open string (Wilson loop) solution in the TsT-deformed AdS5×S5AdS_5\times S_5 spacetime, which are dual to the null polygon Wilson loop and the folded string solution respectively in the final dual geometry.Comment: 24 pages, latex, references added, published versio

    ODE/IM correspondence for modified B2(1)B_2^{(1)} affine Toda field equation

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    We study the massive ODE/IM correspondence for modified B2(1)B_2^{(1)} affine Toda field equation. Based on the ψ\psi-system for the solutions of the associated linear problem, we obtain the Bethe ansatz equations. We also discuss the T-Q relations, the T-system and the Y-system, which are shown to be related to those of the A3/Z2A_3/{\bf Z}_2 integrable system. We consider the case that the solution of the linear problem has a monodromy around the origin, which imposes nontrivial boundary conditions for the T-/Y-system. The high-temperature limit of the T- and Y-system and their monodromy dependence are studied numerically.Comment: 1+21 pages, 2 figures, Typos correcte

    TBA equations and resurgent Quantum Mechanics

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    We derive a system of TBA equations governing the exact WKB periods in one-dimensional Quantum Mechanics with arbitrary polynomial potentials. These equations provide a generalization of the ODE/IM correspondence, and they can be regarded as the solution of a Riemann-Hilbert problem in resurgent Quantum Mechanics formulated by Voros. Our derivation builds upon the solution of similar Riemann-Hilbert problems in the study of BPS spectra in N=2\mathcal{N}=2 gauge theories and of minimal surfaces in AdS. We also show that our TBA equations, combined with exact quantization conditions, provide a powerful method to solve spectral problems in Quantum Mechanics. We illustrate our general analysis with a detailed study of PT-symmetric cubic oscillators and quartic oscillators.Comment: 42 pages, Typos corrected, references are added, published versio

    Limiting Behavior of Constraint Minimizers for Inhomogeneous Fractional Schr\"{o}dinger Equations

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    This paper is devoted to the L2L^2-constraint variational problem \begin{equation*} We study L2L^2-normalized solutions of the following inhomogeneous fractional Schr\"{o}dinger equation \begin{equation*} (-\Delta)^{s} u(x)+V(x)u(x)-a|x|^{-b}|u|^{2\beta^2}u(x)=\mu u(x)\ \ \mbox{in}\ \ \R^{N}. \end{equation*} Here s(12,1)s\in(\frac{1}{2},1), N>2sN>2s, a>0a>0, 0<b<min{N2,1}0<b<\min\{\frac{N}{2},1\}, β=2sbN\beta=\sqrt{\frac{2s-b}{N}} and V(x)0V(x)\geq 0 is an external potential. We get L2L^2-normalized solutions of the above equation by solving the associated constrained minimization problem. We prove that there exists a critical value a>0a^*>0 such that minimizers exist for 0<a<a0<a<a^*, and minimizers do not exist for any a>aa>a^*. In the case of a=aa=a^*, one can obtain the classification results of the existence and non-existence for constraint minimizers, which are depended strongly on the value of V(0)V(0). For V(0)=0V(0)=0, the limiting behavior of nonnegative minimizers is also analyzed when aa tend to aa^* from below

    TTˉT\bar{T} deformation of chiral bosons and Chern-Simons AdS3_3 gravity

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    We study the TTˉT\bar{T} deformation of the chiral bosons and show the equivalence between the chiral bosons of opposite chiralities and the scalar fields at the Hamiltonian level under the deformation. We also derive the deformed Lagrangian of more generic theories which contain an arbitrary number of chiral bosons to all orders. By using these results, we derive the TTˉT\bar{T} deformed boundary action of the AdS3_3 gravity theory in the Chern-Simons formulation. We compute the deformed one-loop torus partition function, which satisfies the TTˉT\bar{T} flow equation up to the one-loop order. Finally, we calculate the deformed stress tensor of a solution describing a BTZ black hole in the boundary theory, which coincides with the boundary stress tensor derived from the BTZ black hole with a finite cutoff.Comment: 29 pages, references adde
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