Related papers: $\tau$-function within group theory approach and i…
In this paper, we construct a new integrable equation which is a generalization of $q$-Toda equation. Meanwhile its soliton solutions are constructed to show its integrable property. Further the Lax pairs of the generalized $q$-Toda…
This paper studies properties of entropy functions that are induced by groups and subgroups. We showed that many information theoretic properties of those group induced entropy functions also have corresponding group theoretic…
One possible way to fix partly a ``canonical definition'' of $\tau$-functions beyond the conventional KP/Toda framework could be to postulate that evolution operators are {\it always} group elements. We discuss implications of this…
The paper is devoted to a description of quantum group structures in the geometric quantization of a self-interacting string field, which appear under a transition from a tree-level of the theory to the account of loop effects in…
This paper begins investigation of the concept of ``generalized $\tau$-function'', defined as a generating function of all the matrix elements of a group element $g \in G$ in a given highest-weight representation of a universal enveloping…
The quantum completion of the space of connections in a manifold can be seen as the set of all morphisms from the groupoid of the edges of the manifold to the (compact) gauge group. This algebraic construction generalizes an analogous…
Exact non-perturbative partition functions of coupling constants and external fields exhibit huge hidden symmetry, reflecting the possibility to change integration variables in the functional integral. In many cases this implies also some…
This review contains a summary of work by J.-L. Gervais and the author on the operator approach to 2d gravity. Special emphasis is placed on the construction of local observables -the Liouville exponentials and the Liouville field itself -…
Quantum integrable systems and their classical counterparts are considered. We show that the symplectic structure and invariant tori of the classical system can be deformed by a quantization parameter $\hbar$ to produce a new (classical)…
It is the goal of this article to extend the notion of quantization from the standard interpretation focused on non-commuting observables defined starting from classical analogues, to the topological equivalents defined in terms of…
Our previous work (math.QA/9808015) introduces the basic notions and announces some results on function theory in the quantum disc. The present paper establishes a relationship between those results and the quantum groups theory.
We introduce the key ideas behind the group field theory approach to quantum gravity, and the basic elements of its formalism. We also briefly report on some recent results obtained in this approach, concerning both the mathematical…
The infinite group of deformed diffeomorphisms of the spacetime continuum is put into the basis of the gauge theory of gravity. This gives rise to some new ways for unification of gravity with other gauge interactions.
This is an introduction to the group field theory approach to quantum gravity, with emphasis on motivations and basic formalism, more than on recent results; we elaborate on the various ingredients, both conceptual and formal, of the…
In this paper, we discuss a group-theoretical generalization of the well-known Gauss formula involving the functionthat counts the number of automorphisms of a finite group. This gives several characterizations of finite cyclic groups.
Foundations of the formal series $*$ -- calculus in deformation quantisation are discussed. Several classes of continuous linear functionals over algebras applied in classical and quantum physics are introduced. The notion of nonnegativity…
The universal Witham hierarchy is considered from the point of view of topological field theories. The $\tau$-function for this hierarchy is defined. It is proved that the algebraic orbits of Whitham hierarchy can be identified with various…
We consider the simplest gauge theories given by one- and two- matrix integrals and concentrate on their stringy and geometric properties. We remind general integrable structure behind the matrix integrals and turn to the geometric…
We review new aspects of integrable systems discovered recently in N=2 supersymmetric gauge theories and their topologically twisted versions. The main topics are (i) an explicit construction of Whitham deformations of the Seiberg-Witten…
The basic elements of the geometric approach to a consistent quantization formalism are summarized, with reference to the methods of the old quantum mechanics and the induced representations theory of Lie groups. A possible relationship…