Kazuhiro Hikami

Note on twisted elliptic genus of K3 surface

Abstract We discuss the possibility of Mathieu group M 24 acting as symmetry group on the K3 elliptic genus as proposed recently by Ooguri, Tachikawa and one of the present authors. One way of testing this proposal is to derive the twisted elliptic genera for all conjugacy classes of M 24 so that we can determine the unique decomposition of expansion coefficients of K3 elliptic genus into irreducible representations of M 24 . In this Letter we obtain all the hitherto unknown twisted elliptic genera and find a strong evidence of Mathieu moonshine.

The AM(1) automata related to crystals of symmetric tensors

A soliton cellular automaton associated with crystals of symmetric tensor representations of the quantum affine algebra Uq′(AM(1)) is introduced. It is a crystal theoretic formulation of the generalized box–ball system in which capacities of boxes and carriers are arbitrary and inhomogeneous. Scattering matrices of two solitons coincide with the combinatorial R matrices of Uq′(AM−1(1)). A piecewise linear evolution equation of the automaton is identified with an ultradiscrete limit of the nonautonomous discrete Kadomtsev–Petviashivili equation. A class of N soliton solutions is obtained through the ultradiscretization of soliton solutions of the latter.A soliton cellular automaton associated with crystals of symmetric tensor representations of the quantum affine algebra Uq′(AM(1)) is introduced. It is a crystal theoretic formulation of the generalized box–ball system in which capacities of boxes and carriers are arbitrary and inhomogeneous. Scattering matrices of two solitons coincide with the combinatorial R matrices of Uq′(AM−1(1)). A piecewise linear evolution equation of the automaton is identified with an ultradiscrete limit of the nonautonomous discrete Kadomtsev–Petviashivili equation. A class of N soliton solutions is obtained through the ultradiscretization of soliton solutions of the latter.

Generalized Volume Conjecture and the A-Polynomials -- the Neumann-Zagier Potential Function as a Classical Limit of Quantum Invariant

Abstract We introduce and study the partition function Z γ ( M ) for the cusped hyperbolic 3-manifold M . We construct formally this partition function based on an oriented ideal triangulation of M by assigning to each tetrahedron the quantum dilogarithm function, which is introduced by Faddeev in his studies of the modular double of the quantum group. Following Thurston and Neumann–Zagier, we deform a complete hyperbolic structure of M , and we define the partition function Z γ ( M u ) correspondingly. This function is shown to give the Neumann–Zagier potential function in the classical limit γ → 0 , and the A -polynomial can be derived from the potential function. We explain our construction by taking examples of 3-manifolds such as complements of hyperbolic knots and a punctured torus bundle over the circle.

Dunkl Operator Formalism for Quantum Many-Body Problems Associated with Classical Root Systems

The integrable quantum many-body systems associated with the classical root systems are formulated in terms of the (trigonometric) Dunkl operators. We define the Dunkl operators by use of the infinite-dimensional representation of the R and K matrices for the Yang-Baxter equation and the reflection equation. The eigenvalues of systems are also given.

Generalized volume conjecture and the A-polynomials: The Neumann-Zagier potential function as a classical limit of the partition function

Abstract We introduce and study the partition function Z γ ( M ) for the cusped hyperbolic 3-manifold M . We construct formally this partition function based on an oriented ideal triangulation of M by assigning to each tetrahedron the quantum dilogarithm function, which is introduced by Faddeev in his studies of the modular double of the quantum group. Following Thurston and Neumann–Zagier, we deform a complete hyperbolic structure of M , and we define the partition function Z γ ( M u ) correspondingly. This function is shown to give the Neumann–Zagier potential function in the classical limit γ → 0 , and the A -polynomial can be derived from the potential function. We explain our construction by taking examples of 3-manifolds such as complements of hyperbolic knots and a punctured torus bundle over the circle.

Yangian symmetry and Virasoro character in a lattice spin system with long-range interactions

Abstract We consider the symmetry of lattice su( n ) spin systems with inverse square exchange; the Haldane-Shastry model and the Polychronakos-Frahm model. It is shown that both systems have yangian symmetry for a finite number of sites. Based on the yangian symmetry the energy spectrum is considered. Furthermore the representation for motifs and the relations with the Virasoro character for su( n ) level-1 Wess-Zumino- Witten theory are discussed.

Infinite Symmetry of the Spin Systems with Inverse Square Interactions

One-dimensional quantum particle system with S U (ν) spins interacting through inverse square interactions is studied. We reveal algebraic structures of the system: hidden symmetry is the \(U(\nu )\cong SU(\nu )\otimes U(1)\) 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, which is the w 1+∞ algebra. With this W algebra we can clarify simultaneously the structures of the Calogero type (1/ x 2 -interactions) and Sutherland type (1/sin 2 x -interactions). The Yangian symmetry is briefly discussed.

Superconformal algebras and mock theta functions

It is known that characters of BPS representations of extended superconformal algebras do not have good modular properties due to extra singular vectors coming from the BPS condition. In order to improve their modular properties we apply the method of Zwegers which has recently been developed to analyze modular properties of mock theta functions. We consider the case of the superconformal algebra at general levels and obtain the decomposition of characters of BPS representations into a sum of simple Jacobi forms and an infinite series of non-BPS representations. We apply our method to study elliptic genera of hyper-Kahler manifolds in higher dimensions. In particular, we determine the elliptic genera in the case of complex four dimensions of the Hilbert scheme of points on K3 surfaces K[2] and complex tori A[[3]].

Quantum integrability of the generalized elliptic Ruijsenaars models

The quantum integrability of the generalized elliptic Ruijsenaars models is shown. These models are mathematically related to the Macdonald operator and the Macdonald - Koornwinder operator, which appeared in the q-orthogonal polynomial theories. We construct these integrable families by using the Yang - Baxter equation and the reflection equation.

Integrability of Calogero-Moser Spin System

One-dimensional quantum particle system with spins is considered. The Hamiltonian of the system (Calogero-Moser spin system) is