Diagonal Ising susceptibility: elliptic integrals, modular forms and Calabi-Yau equations

We give the exact expressions of the partial susceptibilities $\chi^{(3)}_d$ and $\chi^{(4)}_d$ for the diagonal susceptibility of the Ising model in terms of modular forms and Calabi-Yau ODEs, and more specifically, $_3F_2([1/3,2/3,3/2],\, [1,1];\, z)$ and $_4F_3([1/2,1/2,1/2,1/2],\, [1,1,1]; \, z)$ hypergeometric functions. By solving the connection problems we analytically compute the behavior at all finite singular points for $\chi^{(3)}_d$ and $\chi^{(4)}_d$. We also give new results for $\chi^{(5)}_d$. We see in particular, the emergence of a remarkable order-six operator, which is such that its symmetric square has a rational solution. These new exact results indicate that the linear differential operators occurring in the $n$-fold integrals of the Ising model are not only "Derived from Geometry" (globally nilpotent), but actually correspond to "Special Geometry" (homomorphic to their formal adjoint). This raises the question of seeing if these "special geometry" Ising-operators, are "special" ones, reducing, in fact systematically, to (selected, k-balanced, ...) $_{q+1}F_q$ hypergeometric functions, or correspond to the more general solutions of Calabi-Yau equations.Comment: 35 page

The algebraic structure of geometric flows in two dimensions

There is a common description of different intrinsic geometric flows in two dimensions using Toda field equations associated to continual Lie algebras that incorporate the deformation variable t into their system. The Ricci flow admits zero curvature formulation in terms of an infinite dimensional algebra with Cartan operator d/dt. Likewise, the Calabi flow arises as Toda field equation associated to a supercontinual algebra with odd Cartan operator d/d \theta - \theta d/dt. Thus, taking the square root of the Cartan operator allows to connect the two distinct classes of geometric deformations of second and fourth order, respectively. The algebra is also used to construct formal solutions of the Calabi flow in terms of free fields by Backlund transformations, as for the Ricci flow. Some applications of the present framework to the general class of Robinson-Trautman metrics that describe spherical gravitational radiation in vacuum in four space-time dimensions are also discussed. Further iteration of the algorithm allows to construct an infinite hierarchy of higher order geometric flows, which are integrable in two dimensions and they admit immediate generalization to Kahler manifolds in all dimensions. These flows provide examples of more general deformations introduced by Calabi that preserve the Kahler class and minimize the quadratic curvature functional for extremal metrics.Comment: 54 page

Non-commutative NLS-type hierarchies: dressing & solutions

We consider the generalized matrix non-linear Schrodinger (NLS) hierarchy. By employing the universal Darboux-dressing scheme we derive solutions for the hierarchy of integrable PDEs via solutions of the matrix Gelfand-Levitan-Marchenko equation, and we also identify recursion relations that yield the Lax pairs for the whole matrix NLS-type hierarchy. These results are obtained considering either matrix-integral or general $n^{th}$ order matrix-differential operators as Darboux-dressing transformations. In this framework special links with the Airy and Burgers equations are also discussed. The matrix version of the Darboux transform is also examined leading to the non-commutative version of the Riccati equation. The non-commutative Riccati equation is solved and hence suitable conserved quantities are derived. In this context we also discuss the infinite dimensional case of the NLS matrix model as it provides a suitable candidate for a quantum version of the usual NLS model. Similarly, the non-commutitave Riccati equation for the general dressing transform is derived and it is naturally equivalent to the one emerging from the solution of the auxiliary linear problem.Comment: 29 pages, LaTex. Minor modification

The Ising model and Special Geometries

We show that the globally nilpotent G-operators corresponding to the factors of the linear differential operators annihilating the multifold integrals $\chi^{(n)}$ of the magnetic susceptibility of the Ising model ($n \le 6$) are homomorphic to their adjoint. This property of being self-adjoint up to operator homomorphisms, is equivalent to the fact that their symmetric square, or their exterior square, have rational solutions. The differential Galois groups are in the special orthogonal, or symplectic, groups. This self-adjoint (up to operator equivalence) property means that the factor operators we already know to be Derived from Geometry, are special globally nilpotent operators: they correspond to "Special Geometries". Beyond the small order factor operators (occurring in the linear differential operators associated with $\chi^{(5)}$ and $\chi^{(6)}$), and, in particular, those associated with modular forms, we focus on the quite large order-twelve and order-23 operators. We show that the order-twelve operator has an exterior square which annihilates a rational solution. Then, its differential Galois group is in the symplectic group $Sp(12, \mathbb{C})$. The order-23 operator is shown to factorize in an order-two operator and an order-21 operator. The symmetric square of this order-21 operator has a rational solution. Its differential Galois group is, thus, in the orthogonal group $SO(21, \mathbb{C})$.Comment: 33 page

Solving Virasoro Constraints in Matrix Models

This is a brief review of recent progress in constructing solutions to the matrix model Virasoro equations. These equations are parameterized by a degree n polynomial W_n(x), and the general solution is labeled by an arbitrary function of n-1 coefficients of the polynomial. We also discuss in this general framework a special class of (multi-cut) solutions recently studied in the context of \cal N=1 supersymmetric gauge theories.Comment: 9 pages, LaTeX, contribution to the 37th International Symposium Ahrenshoop on the Theory of Elementary Particle