Quantum Sine(h)-Gordon Model and Classical Integrable Equations

We study a family of classical solutions of modified sinh-Gordon equation, $\partial_z\partial_{{\bar z}} \eta-\re^{2\eta}+p(z)\,p({\bar z})\ \re^{-2\eta}=0$with$p(z)=z^{2\alpha}-s^{2\alpha}$. We show that certain connection coefficients for solutions of the associated linear problem coincide with the$Q$-function of the quantum sine-Gordon$(\alpha>0)$or sinh-Gordon$(\alpha<-1)$ models.Comment: 35 pages, 3 figure

g-Functions and gluon scattering amplitudes at strong coupling

We study gluon scattering amplitudes/Wilson loops in N=4 super Yang-Mills theory at strong coupling by calculating the area of the minimal surfaces in AdS_3 based on the associated thermodynamic Bethe ansatz system. The remainder function of the amplitudes is computed by evaluating the free energy, the T- and Y-functions of the homogeneous sine-Gordon model. Using conformal field theory (CFT) perturbation, we examine the mass corrections to the free energy around the CFT point corresponding to the regular polygonal Wilson loop. Based on the equivalence between the T-functions and the g-functions, which measure the boundary entropy, we calculate corrections to the T- and Y-functions as well as express them at the CFT point by the modular S-matrix. We evaluate the remainder function around the CFT point for 8 and 10-point amplitudes explicitly and compare these analytic expressions with the 2-loop formulas. The two rescaled remainder functions show very similar power series structures.Comment: 51 pages, 4 figures, v2: some comments and references added, based on the published version, v3: minor change

On the classical equivalence of monodromy matrices in squashed sigma model

We proceed to study the hybrid integrable structure in two-dimensional non-linear sigma models with target space three-dimensional squashed spheres. A quantum affine algebra and a pair of Yangian algebras are realized in the sigma models and, according to them, there are two descriptions to describe the classical dynamics 1) the trigonometric description and 2) the rational description, respectively. For every description, a Lax pair is constructed and the associated monodromy matrix is also constructed. In this paper we show the gauge-equivalence of the monodromy matrices in the trigonometric and rational description under a certain relation between spectral parameters and the rescalings of sl(2) generators.Comment: 32pages, 3figures, references added, introduction and discussion sections revise

Thermodynamic Bethe Ansatz Equations for Minimal Surfaces in AdS_3

We study classical open string solutions with a null polygonal boundary in AdS_3 in relation to gluon scattering amplitudes in N=4 super Yang-Mills at strong coupling. We derive in full detail the set of integral equations governing the decagonal and the dodecagonal solutions and identify them with the thermodynamic Bethe ansatz equations of the homogeneous sine-Gordon models. By evaluating the free energy in the conformal limit we compute the central charges, from which we observe general correspondence between the polygonal solutions in AdS_n and generalized parafermions.Comment: 25 pages, 4 figures, v2: a figure and references added, minor corrections, v3: references added, minor corrections, to appear in JHE

Exact Results on the ABJM Fermi Gas

We study the Fermi gas quantum mechanics associated to the ABJM matrix model. We develop the method to compute the grand partition function of the ABJM theory, and compute exactly the partition function Z(N) up to N=9 when the Chern-Simons level k=1. We find that the eigenvalue problem of this quantum mechanical system is reduced to the diagonalization of a certain Hankel matrix. In reducing the number of integrations by commuting coordinates and momenta, we find an exact relation concerning the grand partition function, which is interesting on its own right and very helpful for determining the partition function. We also study the TBA-type integral equations that allow us to compute the grand partition function numerically. Surprisingly, all of our exact results of the partition functions are written in terms of polynomials of 1/pi with rational coefficients.Comment: 41 pages, 4 figure

Six-dimensional supersymmetric gauge theories, quantum cohomology of instanton moduli spaces and gl(N) Quantum Intermediate Long Wave Hydrodynamics

We show that the exact partition function of U(N) six-dimensional gauge theory with eight supercharges on \u21022 7 S 2 provides the quantization of the integrable system of hydrodynamic type known as gl(N) periodic Intermediate Long Wave (ILW). We characterize this system as the hydrodynamic limit of elliptic Calogero-Moser integrable system. We compute the Bethe equations from the effective gauged linear sigma model on S 2 with target space the ADHM instanton moduli space, whose mirror computes the Yang-Yang function of gl(N) ILW. The quantum Hamiltonians are given by the local chiral ring observables of the six-dimensional gauge theory. As particular cases, these provide the gl(N) Benjamin-Ono and Korteweg-de Vries quantum Hamiltonians. In the four dimensional limit, we identify the local chiral ring observables with the conserved charges of Heisenberg plus W N algebrae, thus providing a gauge theoretical proof of AGT correspondence. \ua9 2014 The Author(s)