Takeshi Oota

Method of generating q-expansion coefficients for conformal block and N=2 Nekrasov function by β-deformed matrix model

Abstract We observe that, at β -deformed matrix models for the four-point conformal block, the point q = 0 is the point where the three-Penner type model becomes a pair of decoupled two-Penner type models and where, in the planar limit, (an array of), two-cut eigenvalue distribution(s) coalesce into (that of) one-cut one(s). We treat the Dotsenko–Fateev multiple integral, with their paths under the recent discussion, as perturbed double-Selberg matrix model (at q = 0 , it becomes a pair of Selberg integrals) to construct two kinds of generating functions for the q -expansion coefficients and compute some. A formula associated with the Jack polynomial is noted. The second Nekrasov coefficient for SU ( 2 ) with N f = 4 is derived. A pair of Young diagrams appears naturally. The finite N loop equation at q = 0 as well as its planar limit is solved exactly, providing a useful tool to evaluate the coefficients as those of the resolvents. The planar free energy in the q -expansion is computed to the lowest non-trivial order. A free field representation of the Nekrasov function is given.

Closed conformal Killing–Yano tensor and Kerr–NUT–de Sitter space–time uniqueness

We study spacetimes with a closed conformal Killing-Yano tensor. It is shown that the D-dimensional Kerr-NUT-de Sitter spacetime constructed by Chen-Lu- Pope is the only spacetime admitting a rank-2 closed conformal Killing-Yano tensor with a certain symmetry.

Toric Sasaki–Einstein manifolds and Heun equations

Abstract Symplectic potentials are presented for a wide class of five-dimensional toric Sasaki–Einstein manifolds, including L a , b , c which was recently constructed by Cvetic et al. The spectrum of the scalar Laplacian on L a , b , c is also studied. The eigenvalue problem leads to two Heuns differential equations and the exponents at regular singularities are directly related to the toric data. By combining knowledge of the explicit symplectic potential and the exponents, we show that the ground states, or equivalently holomorphic functions, have one-to-one correspondence with the integral lattice points in the convex polyhedral cone. The scaling dimensions of the holomorphic functions are simply given by the scalar products of the Reeb vector and the integral vectors, which are consistent with R -charges of the BPS states in the dual quiver gauge theories.

Separability of Dirac equation in higher dimensional Kerr–NUT–de Sitter spacetime

Abstract It is shown that the Dirac equations in general higher dimensional Kerr–NUT–de Sitter spacetimes are separated into ordinary differential equations.

Explicit toric metric on resolved Calabi–Yau cone

Abstract We present an explicit non-singular complete toric Calabi–Yau metric using the local solution recently found by Chen, Lu and Pope. This metric gives a new supergravity solution representing D3-branes.

Closed conformal Killing-Yano tensor and geodesic integrability

Assuming the existence of a single rank-2 closed conformal Killing–Yano tensor with a certain symmetry we show that there exists mutually commuting rank-2 Killing tensors and Killing vectors. We also discuss the condition of separation of variables for the geodesic Hamilton–Jacobi equations.

Kerr–NUT–de Sitter curvature in all dimensions

In slotting tools with exchangeable cutting inserts the insert is adapted to be clamped between two narrow jaws situated exactly above each other in the front end of a holder the lower jaw of which is formed integrally with the holder and defines the cutting seat. The upper jaw, the front portion of which reaches outside the principal part of the holder, consists of a loose member which is adapted to be inserted in a recess in the principal part of the holder, behind the cutting seat of the lower jaw, and is locked against lateral displacement. This loose member is retained in its position in the recess exclusively because of the tightening effect provided between the jaws over the clamped cutting insert.

2d–4d connection between q-Virasoro/W block at root of unity limit and instanton partition function on ALE space

Abstract We propose and demonstrate a limiting procedure in which, starting from the q -lifted version (or K -theoretic five-dimensional version) of the (W)AGT conjecture to be assumed in this paper, the Virasoro/ W block is generated in the r -th root of unity limit in q in the 2d side, while the same limit automatically generates the projection of the five-dimensional instanton partition function onto that on the ALE space R 4 / Z r . This circumvents case-by-case conjectures to be made in a wealth of examples found so far. In the 2d side, we successfully generate the super-Virasoro algebra and the proper screening charge in the q → − 1 , t → − 1 limit, from the defining relation of the q -Virasoro algebra and the q -deformed Heisenberg algebra. The central charge obtained coincides with that of the minimal series carrying odd integers of the N = 1 superconformal algebra. In the r -th root of unity limit in q in the 2d side, we give some evidence of the appearance of the parafermion-like currents. Exploiting the q -analysis literatures, q -deformed su ( n ) block is readily generated both at generic q , t and the r -th root of unity limit. In the 4d side, we derive the proper normalization function for general ( n , r ) that accomplishes the automatic projection through the limit.

Generalized Kerr-NUT-de Sitter metrics in all dimensions

We classify all spacetimes with a closed rank-2 conformal Killing-Yano tensor. They give a generalization of Kerr-NUT-de Sitter spacetimes. The Einstein condition is explicitly solved and written as an indefinite integral. It is characterized by a polynomial in the integrand. We briefly discuss the smoothness conditions of the Einstein metrics over compact Riemannian manifolds.

SEPARABILITY OF GRAVITATIONAL PERTURBATION IN GENERALIZED KERR–NUT–DE SITTER SPACE–TIME

Generalized Kerr–NUT–de Sitter space–time is the most general space–time which admits a rank-2 closed conformal Killing–Yano tensor. It contains the higher-dimensional Kerr–de Sitter black holes with partially equal angular momenta. We study the separability of gravitational perturbations in the generalized Kerr–NUT–de Sitter space–time. We show that a certain type of tensor perturbations admits the separation of variables. The linearized perturbation equations for the Einstein condition are transformed into the ordinary differential equations of Fuchs type.