Entropy Product Function and Central charges in NUT Geometry

We define an entropy product function (EPF) for Taub-Newman-Unti-Tamburino(TNUT) black hole(BH) following the prescription suggested by Wu et al. [PRD 100, 101501(R) (2019)]. The prescription argues that a generic four-dimensional TNUT spacetime might be expressed in terms of three or four different types of thermodynamic hairs. They can be defined as the Komar mass(M = m), the angular momentum(Jn = mn), the gravitomagnetic charge (N = n), the dual (magnetic) mass (M̃ = n). Taking this prescription and using the EPF, we derive the central charges of dual CFT (conformal field theory) via Cardy’s formula. Remarkably, we find that for TNUT BH there exists a relation between the central charges and EPF as c = 6 ( ∂calF ∂calNi ) , where calF is EPF and calNi is one of the integer-valued charges i.e. the NUT charges(N) or any new conserved charges(JN ). We reverify these results by calculating the exact values of different thermodynamic parameters. We define the EPF calF from the first law of thermodynamics of both horizons. Moreover, we write the first laws of both the horizons for left-moving and right-moving sectors. Introducing the B\’{e}zout’s identity, we show that for TNUT BH one can generate more holographic descriptions described by a pair of integers (a, b). More holographic pictures have a great significance in understanding the holographic nature of quantum gravity. Furthermore, using the EPF we derive the central charges for Reissner-Nordstr\”{o}m-NUT(RNNUT) BH, Kerr-Taub- NUT~(KNUT) BH and Kerr-Newman-NUT~(KNNUT) BH proved that they are equal in both sectors provided that the EPF is mass-independent(or universal).