Related papers: A new Lax pair for the sixth Painlev\'e equation a…

In families of Painlev\'e VI differential equations having common algebraic solutions we classify all the members which come from geometry, i.e. the corresponding linear differential equations which are Picard-Fuchs associated to families…

For each Painlev\'e system P_J except the first one, we have a B\"acklund transformation group which is a lift of an affine Weyl group. In this paper, we show that the B\"acklund transformation groups for J=V,IV,III,II are successively…

It is shown that a generalization of the Painlev\'e-II equation (P-II) to a system of coupled equations with symmetry breaking terms is integrable. A Lax pair for this system is used to relate the asymptotic behavior of the solutions at…

We give a survey of the following six closely related topics: (i) a general method for constructing a soliton hierarchy from a splitting of a loop algebra into positive and negative subalgebras, together with a sequence of commuting…

The group reduction procedure is applied to vector generalizations of the NLS, mKdV, and KdV equations. The resulting ODE systems admit isomonodromic Lax representations and are multicomponent generalizations of the Painlev\'e equations…

In this note we discuss the classification of duality orbits of N = 8 gauged supergravity models. Using tensor classifiers, we show that there is a one-parameter family of inequivalent SO(8) gauged supergravity theories. We briefly discuss…

Left invariant affine structures in a Lie group $G$ are in one-to-one correspondence with left-symmetric algebras over its Lie algebra $\mathfrak g=T_eG$ (``over'' means that the commutator $[x,y]=xy-yx$ coincides with the Lie bracket;…

Pulling back the weight system associated with the spinor representation of the Lie algebra so(7) by the universal Vassiliev-Kontsevich invariant yields a numerical link invariant with values in formal power series. Computing some skein…

Equivalence is established between a special class of Painleve VI equations parametrized by a conformal dimension $\mu$, time dependent Euler top equations, isomonodromic deformations and three-dimensional Frobenius manifolds. The…

New gaugings of four dimensional N=8 supergravity are constructed, including one which has a Minkowski space vacuum that preserves N=2 supersymmetry and in which the gauge group is broken to $SU(3)xU(1)^2$. Previous gaugings used the form…

We consider a 6-dimensional supersymmetric SU(6) gauge theory and compactify two extra-dimensions on a multiply-connected manifold with non-trivial topology. The SU(6) is broken down to the Standard Model gauge groups in two steps by an…

In this paper, we propose a 3-parameter family of coupled Painlev\'e III systems in dimension four with affine Weyl group symmetry of type $A_5^{(2)}$. We also propose its symmetric form in which the $A_5^{(2)}$-symmetries become clearly…

Isomonodromy for the fifth Painlev\'e equation ${\rm P}_5$ is studied in detail in the context of certain moduli spaces for connections, monodromy, the Riemann-Hilbert morphism, and Okamoto-Painlev\'e spaces. This involves explicit formulas…

We present the bilinear forms of the (continuous) Painlev\'e equations obtained from the continuous limit of the analogous expresssions for the discrete ones. The advantage of this method is that it leads to very symmetrical results. A new…

A Lie pair is an inclusion $A$ to $L$ of Lie algebroids over the same base manifold. In an earlier work, the third author with Bandiera, Sti\'{e}non, and Xu introduced a canonical $L_{\leqslant 3}$ algebra $\Gamma(\wedge^\bullet A^\vee…

Inspired by Etingof--Varchenko's dynamical fusion, dynamical $R$-matrix, and dynamical Weyl group for Lie algebras, we introduce, for split symmetric pairs, versions of dynamical fusion, dynamical $K$-matrix, and dynamical Weyl group. We…

We show that symmetric polynomials previously introduced by the author satisfy a certain differential equation. After a change of variables, it can be written as a non-stationary Schr\"odinger equation with elliptic potential, which is…

We find and study a five-parameter family of four-dimensional coupled Painlev\'e V systems with affine Weyl group symmetry of type $D_5^{(1)}$. We then give an explicit description of a confluence from those systems to a four-parameter…

We modify the classical Paley-Wiener spaces $PW_x$ of entire functions of finite exponential type at most $x>0$, which are square integrable on the real line, via the additional condition of vanishing at finitely many complex points $z_1,…

We further explore the correspondence between N=2 supersymmetric SU(2) gauge theory with four flavors on epsilon-deformed backgrounds and conformal field theory, with an emphasis on the epsilon-expansion of the partition function natural…