The SU(v) Calogero spin system

A one-dimensional quantum particle system in which particles with su(v) spins interact through inverse square interactions is introduced. We refer to it as the SU(v) Calogero spin system. Using the quantum inverse scattering method, we reveal algebraic structures of the system: hidden symmetry is the U(v) − SU(v) U(1) current algebra. This is consistent with the fact that the ground-state wave function is a solution of the Knizhnik-Zamolodchikov equation. Furthermore we show that the system has a higher symmetry, known as the w_{1 + ∞}-algebra. With this W-algebra we have a unified viewpoint on the integrable quantum particle systems with long-range interactions such as the Calogero type (1/x²-interactions) and Sutherland type (1/sin²x-interactions). The Yangian symmetry is briefly discussed.     

$${\mathcal{N}=2}$$ Superconformal Algebra and the Entropy of Calabi–Yau Manifolds

We use the representation theory of \({\mathcal{N}=2}\) superconformal algebra to study the elliptic genera of Calabi–Yau (CY) D-folds. We compute the entropy of CY manifolds from the growth rate of multiplicities of the massive (non-BPS) representations in the decomposition of their elliptic genera. We find that the entropy of CY manifolds of complex dimension D behaves differently depending on whether D is even or odd. When D is odd, CY entropy coincides with the entropy of the corresponding hyperKähler (D ? 3)-folds due to a structural theorem on Jacobi forms. In particular, we find that the Calabi–Yau 3-fold has a vanishing entropy. At D > 3, using our previous results on hyperKähler manifolds, we find \({S_{CY_D}\sim 2\pi \sqrt{\frac{(D-3)^2}{2(D-1)}n}}\). When D is even, we find the behavior of CY entropy behaving as \({S_{CY_D}\sim 2 \pi\sqrt{\frac{D-1}{2}n}}\). These agree with Cardy’s formula at large D.     

Three-loop ß-functions of non-linear σ models on symmetric spaces

It is shown that the renormalization group ß-functions of the two-dimensional non-linear σ models on various symmetric spaces are determined up to three-loop order by the isomorphic relations between the classical groups.     

Affine R-Matrix and the Generalized Elliptic Ruijsenaars Models

Commutative elliptic difference operators associated with the affine root systems are constructed in terms of affine R-matrices. These operators describe the Ruijsenaars models with elliptic potentials and reduce to the Macdonald operators in the trigonometric limit.     

Isomorphism and the β-function of the non-linear σ model in symmetric spaces

The renormalization group β-function of the non-linear σ model in symmetric spaces is discussed via the isomorphic relation and the reciprocal relation about a parameter α. The four-loop term is investigated and the symmetric properties of the β-function are studied. The four-loop term in the β-function is shown to be vanishing for the orthogonal Anderson localization problem.     

On the Quantum Invariants for the Spherical Seifert Manifolds

We study the Witten–Reshetikhin–Turaev SU(2) invariant for the Seifert manifolds S ³/Gamma where Γ is a finite subgroup of SU(2). We show that the WRT invariants can be written in terms of the Eichler integral of modular forms with half-integral weight, and we give an exact asymptotic expansion of the invariants by use of the nearly modular property of the Eichler integral. We further discuss that those modular forms have a direct connection with the polyhedral group by showing that the invariant polynomials of modular forms satisfy the polyhedral equations associated to Γ.     

Quantum effect,critical dynamics and one-particle excitation in 1/n expansion

By quantizing Ma's Hamiltonian, quantum effect on η, the energy spectrum of one-particle excitation and the dynamic scaling law are studied up to O(1/n). The case just at the critical point is considered.