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

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

Recently, the separability of Klein-Gordon equations in higher dimensional Kerr-NUT-de Sitter spacetimes [1] was shown by Frolov, Krtouš and Kubizňák [2]. This separation is deeply related to that of geodesic Hamilton-Jacobi equations. Indeed, a geometrical object called conformal Killing-Yano tensor plays an important role in the separability theory [3,4,5,2,6,7,8,9]. However, at present, a similar separation of the variables of Dirac equations is lacking, although the separability in the four dimensional Kerr geometry was given by Chandrasekhar [10]. In this paper we shall show that Dirac equations can also be separated in general Kerr-NUT-de Sitter spacetimes.
The D-dimensional Kerr-NUT-de Sitter metrics are written as follows [1]: (a) D = 2n The functions Q µ (µ = 1, 2, · · · , n) are given by where X µ is a function depending only on the coordinate x µ , and A (k) and A (k) µ are the elementary symmetric functions of {x 2 ν } and {x 2 ν } ν =µ respectively: The metrics are Einstein if X µ takes the form [1,11] (a) D = 2n where c, c 2k and b µ are free parameters.

D=2n
For the metric (1) we introduce the following orthonormal basis {e a } = {e µ , e n+µ } (µ = 1, 2, · · · , n): The dual vector fields are given by The spin connection is calculated as [11] Then, the Dirac equation is written in the form where D a is a covariant differentiation, From (9), (10) and (12), we obtain the explicit expression for the Dirac equation Let us use the following representation of γ-matrices: {γ a , γ b } = 2δ ab , where I is the 2 × 2 identity matrix and σ i are the Pauli matrices. In this representation, we write the 2 n components of the spinor field as Ψ ǫ 1 ǫ 2 ···ǫn (ǫ µ = ±1), and it follows that By the isometry the spinor field takes the form with arbitrary constants N k . Substituting (15) into (13), we obtain where we have introduced the function which depends only on x µ . Consider now the region x µ − x ν > 0 for µ < ν and x µ + x ν > 0. Let us define Then, one can obtain an equality . (20) Now we show that the Dirac equation allows a separation of variables by settinĝ It should be noticed that By using (20) and (22), the substitution of (21) into (17) leads to n µ=1 P (µ) where P

(µ)
ǫµ is a function of the coordinate x µ only, Putting with arbitrary constants q j , we find Thus, the functions χ (µ) ǫµ satisfy the ordinary differential equations
We have shown the separation of variables of Dirac equations in general Kerr-NUT-de Sitter spacetimes. An interesting problem is to investigate the origin of separability. In the case of geodesic Hamilton-Jacobi equations and Klein-Gordon equations we know that the existence of separable coordinates comes from that of a rank-2 closed conformal Killing-Yano tensor. However, we have no clear answer for Dirac equations. As another problem we can study eigenvalues of Dirac operators on Sasaki-Einstein manifolds. Indeed, as shown in [1,12,13,14], the BPS limit of odd-dimensional Kerr-NUT-de Sitter metrics leads to Sasaki-Einstein metrics. Especially, the five-dimensional metrics are important from the point of view of AdS/CFT correspondence.