A spherical charged black hole in thermal equilibrium is considered from the perspective of a grand canonical ensemble which electrostatic potential, temperature, and surface area are specified at finite boundary. correspondence established between boundary-value data well-posed problem region Euclidean spacetime freely chosen thermodynamic specifying ensemble. The Hamiltonian Gauss's-law constraints solved eliminated Einstein-Maxwell action, producing "reduced action" that depends upon two...

We give necessary and sufficient conditions for the matrix equations ATX\pm XTA=B to admit solutions their general solution. Because A may be singular, solution involves generalized inverse of A. The (-) equation underlies modern R-matrix approach completely integrable Hamiltonian systems. This paper provides a new algorithmic constructing such R-matrices.

Using a previously developed formulation of black-hole thermodynamics for system finite size, we show that the partition function related canonical ensemble can be used to obtain density states in microcanonical ensemble. We work with zero-loop approximation based on classical solutions fixed boundary data and corresponding by using an inverse Laplace transform. The computation requires introduction uniformizing variable so path defined along which action stable black hole is single valued...

We construct a two parameter family of 2-particle Hamiltonians closed under the duality operation interchanging (relative) momentum and coordinate. Both coordinate dependence are elliptic, modulus torus is non-trivial function This model contains as limiting cases standard Ruijsenaars-Calogero Toda Hamiltonians, which at most elliptic in coordinates, but not momenta.

Some new techniques are developed and expansions given for field theories on ${S}^{1}\ifmmode\times\else\texttimes\fi{}\ensuremath{\Sigma}$. These particularly relevant theory at finite temperature or twisted-field calculations. Application is made to the effective potential high-temperature regime of common relativistic phase-space integrals.

The general analytic solution to the functional equation $$ \phi_1(x+y)= { \biggl|\matrix{\phi_2(x)&\phi_2(y)\cr\phi_3(x)&\phi_3(y)\cr}\biggr|} \over \biggl|\matrix{\phi_4(x)&\phi_4(y)\cr\phi_5(x)&\phi_5(y)\cr}\biggr|} } is characterized. Up action of symmetry group, this described in terms Weierstrass elliptic functions. We illustrate our theory by applying it classical addition theorems Jacobi functions and equations \phi_1(x+y)=\phi_4(x)\phi_5(y)+\phi_4(y)\phi_5(x) \[ \Psi _1(x+y)=\Psi...

An old conjecture claims that commuting Hamiltonians of the double-elliptic integrable system are constructed from theta-functions associated with Riemann surfaces Seiberg-Witten family, moduli treated as dynamical variables and differential providing pre-symplectic structure. We describe a number theta-constant equations needed to prove this for N-particle system. These provide an alternative method derive prepotential we illustrate by calculating perturbative contribution. evidence...

We seek to clarify some of the physical aspects Ruijsenaars-Schneider models. This important class models was presented as a relativistic generalisation Calogero-Moser (i.e. integrable particle dynamics with long range interactions on line) but, we shall argue, this description is misleading. It far better simply view one-parameter By viewing describing certain eigenvalue motions can appreciate generic nature

The Somos 4 sequences are a family of satisfying fourth order bilinear recurrence relation. In recent work, one us has proved that the general term in such can be expressed terms Weierstrass sigma function for an associated elliptic curve. Here we derive analogous with hyperelliptic curve genus two defined by affine model $y^2=4x^5+c_4 x^4+ ... +c_1 x+c_0$. We show curves satisfy recurrences 8. proof requires addition formula which involves Kleinian its argument shifted Abelian image reduced...