Extended phase space thermodynamics for third order Lovelock black holes in diverse dimensions

Treating the cosmological constant as thermodynamic pressure and its conjugate as thermodynamic volume, we investigate the critical behavior of the third order Lovelock black holes in diverse dimensions. For black hole horizons with different normalized sectional curvature $k=0,\pm1$, the corresponding critical behaviors differ drastically. For $k=0$, there is no critical point in the extended thermodynamic phase space. For $k=-1$, there is a single critical point in any dimension $d\geq 7$, and for $k=+1$, there is a single critical point in $7$ dimension and two critical points in $8,9,10,11$ dimensions. We studied the corresponding phase structures in all possible cases.Comment: pdflatex, 22 pages, 36 eps figures included. V2: minor corrections and new reference

Gauss-Bonnet coupling constant as a free thermodynamical variable and the associated criticality

The thermodynamic phase space of Gauss-Bonnet (GB) AdS black holes is extended, taking the inverse of the GB coupling constant as a new thermodynamic pressure $P_{\mathrm{GB}}$. We studied the critical behavior associated with $P_{\mathrm{GB}}$ in the extended thermodynamic phase space at fixed cosmological constant and electric charge. The result shows that when the black holes are neutral, the associated critical points can only exist in five dimensional GB-AdS black holes with spherical topology, and the corresponding critical exponents are identical to those for Van der Waals system. For charged GB-AdS black holes, it is shown that there can be only one critical point in five dimensions (for black holes with either spherical or hyperbolic topologies), which also requires the electric charge to be bounded within some appropriate range; while in $d>5$ dimensions, there can be up to two different critical points at the same electric charge, and the phase transition can occur only at temperatures which are not in between the two critical values.Comment: 23 pages. V2: modified all P_{GB}-r_+ plots using dimensionless variables, added comments on the relationship to Einstein limi

Generalized Projective Representations for sl(n+1)

It is well known that $n$-dimensional projective group gives rise to a non-homogenous representation of the Lie algebra $sl(n+1)$ on the polynomial functions of the projective space. Using Shen's mixed product for Witt algebras (also known as Larsson functor), we generalize the above representation of $sl(n+1)$ to a non-homogenous representation on the tensor space of any finite-dimensional irreducible $gl(n)$-module with the polynomial space. Moreover, the structure of such a representation is completely determined by employing projection operator techniques and well-known Kostant's characteristic identities for certain matrices with entries in the universal enveloping algebra. In particular, we obtain a new one parameter family of infinite-dimensional irreducible $sl(n+1)$-modules, which are in general not highest-weight type, for any given finite-dimensional irreducible $sl(n)$-module. The results could also be used to study the quantum field theory with the projective group as the symmetry.Comment: 24page