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We present Painlev{e} VI sigma form equations for the general Ising low and high temperature two-point correlation functions $ C(M,N)$ with $M leq N $ in the special case $ u = -k$ where $ u = , sinh 2E_h/k_BT/sinh 2E_v/k_BT$. More specifically four different non-linear ODEs depending explicitly on the two integers $M $ and $N$ emerge: these four non-linear ODEs correspond to distinguish respectively low and high temperature, together with $ M+N$ even or odd. These four different non-linear ODEs are also valid for $M ge N$ when $ u = -1/k$. For the low-temperature row correlation functions $ C(0,N)$ with $ N$ odd, we exhibit again for this selected $ u = , -k$ condition, a remarkable phenomenon of a Painleve VI sigma function being the sum of four Painleve VI sigma functions having the same Okamoto parameters. We show in this $ u = , -k$ case for $ T < T_c $ and also $ T > T_c$, that $ C(M,N)$ with $ M leq N $ is given as an $ N times N$ Toeplitz determinant.
We present an approach that gives rigorous construction of a class of crossing invariant functions in $c=1$ CFTs from the weakly invariant distributions on the moduli space $mathcal M_{0,4}^{SL(2,mathbb{C})}$ of $SL(2,mathbb{C})$ flat connections on
In this paper we reconsider the problem of the Euler parametrization for the unitary groups. After constructing the generic group element in terms of generalized angles, we compute the invariant measure on SU(N) and then we determine the full range o
In previous papers, a geometric framework has been developed to describe non-conservative field theories as a kind of modified Lagrangian and Hamiltonian field theories. This approach is that of $k$-contact Hamiltonian systems, which is based on the
A universal C*-algebra of the electromagnetic field is constructed. It is represented in any quantum field theory which incorporates electromagnetism and expresses basic features of this field such as Maxwells equations, Poincare covariance and Einst
We develop a group-theoretical approach to describe $N$-component composite bosons as planar electrons attached to an odd number $f$ of Chern-Simons flux quanta. This picture arises when writing the Coulomb exchange interaction as a quantum Hall ferr