LIE THEORY AND ITS APPLICATIONS IN PHYSICS: VIII International Workshop

Integrable Conformal Field Theory—A Case Study
View Description Hide DescriptionOver the last decades, 2‐dimensional conformal field theory has been developed into a powerful tool that has been applied to problems in diverse branches of physics and mathematics. Models are usually solved algebraically by exploiting certain infinite dimensional symmetries. But the presence of sufficient world‐sheet symmetry is a rather exceptional feature, one that is e.g. not present for curved string backgrounds at generic points in moduli space. In this note I review some recent work which aims at computing spectra of conformal sigma models without spectrum generating symmetries. Our main results are illustrated at the example of complex projective superspace This note is based on several publications with C. Candu, T. Creutzig, V. Mitev, T. Quella and H. Saleur.

Minimal Representations and Reductive Dual Pairs in Conformal Field Theory
View Description Hide DescriptionA minimal representation of a simple non‐compact Lie group is obtained by “quantizing” the minimal nilpotent coadjoint orbit of its Lie algebra. It provides context for Roger Howe’s notion of a reductive dual pair encountered recently in the description of global gauge symmetry of a (4‐dimensional) conformal observable algebra. We give a pedagogical introduction to these notions and point out that physicists have been using both minimal representations and dual pairs without naming them and hence stand a chance to understand their theory and to profit from it.

4‐Point Superconformal Blocks in SCFT
View Description Hide DescriptionThe recursive relations for the 4‐point superconformal blocks in SCFT are discussed. The case of the blocks corresponding to correlation functions of four Ramond fields factorized on NS states is presented in detail.

On p‐Adic Sector of Open Scalar Strings and Zeta Field Theory
View Description Hide DescriptionWe consider construction of Lagrangians which may be suitable for description of p‐adic sector of an open scalar string. Such Lagrangians have their origin in Lagrangian for a single p‐adic string and they contain the Riemann zeta function with the d’Alembertian in its argument. However, investigation of the field theory with Riemann zeta function is interesting in itself as well. We present a brief review and some new results.

Operators on Pure Spinor Spaces
View Description Hide DescriptionPure spinors are relevant to the formulation of supersymmetric theories, and provide the only known way to maintain manifest maximal supersymmetry. The (non‐linear) pure spinor constraint makes it nontrivial to find well defined operators on pure spinor wave functions. We discuss how such operators are defined. One application concerns covariant gauge fixing in maximally supersymmetric Yang‐Mills (and string theory). Another issue is the construction of a manifestly supersymmetric action for 11 ‐dimensional supergravity in terms of a scalar superfield. We describe some work in progress.

Asymmetric Wormholes via Electrically Charged Lightlike Branes
View Description Hide DescriptionWe consider a self‐consistent Einstein‐Maxwell‐Kalb‐Ramond system in the bulk space‐time interacting with a variable‐tension electrically charged lightlike brane . The latter serves both as a material and charge source for gravity and electromagnetism, as well as it dynamically generates a bulk space varying cosmological constant. We find an asymmetric wormhole solution describing two “universes” with different spherically symmetric black‐hole‐type geometries connected through a “throat” occupied by the lightlike brane. The electrically neutral “left universe” comprises the exterior region of Schwarzschild‐de‐Sitter (or pure Schwarzschild) space‐time above the inner (Schwarzschild‐type) horizon, whereas the electrically charged “right universe” consists of the exterior Reissner‐Nordström (or Reissner‐Nordström‐de‐Sitter) black hole region beyond the outer Reissner‐Nordström horizon. All physical parameters of the wormhole are uniquely determined by two free parameters—the electric charge and Kalb‐Ramond coupling of the lightlike brane.

Tensor Models as Theory of Dynamical Fuzzy Spaces and General Relativity
View Description Hide DescriptionThe tensor model is discussed as theory of dynamical fuzzy spaces in order to formulate gravity on fuzzy spaces. The numerical analyses of the tensor models possessing Gaussian background solutions have shown that the low‐lying long‐wavelength fluctuations around the backgrounds are in remarkable agreement with the geometric fluctuations on flat spaces in the general relativity. It has also been shown that part of the orthogonal symmetry of the tensor model spontaneously broken by the backgrounds agrees with the local translation symmetry of the general relativity. Thus the tensor model provides an interesting model of simultaneous emergence of space, the general relativity, and its local translation symmetry.

EMP Reformulations of Einstein’s Equations as an Application of a Property of Suitable Second Order Differential Equations
View Description Hide DescriptionRecently various authors have shown that Einstein’s gravitational field equations for a number of scalar field cosmological models can be equivalently reformulated in terms of a generalized Ermakov‐Milne‐Pinney (EMP) type equation. We will show how these correspondences can be realized as an application of a general property of second order differential equations, in which solutions of under a suitable mapping, transform to solutions of the generalized EMP equation

Analytically Solvable Quantum Hamiltonians and Relations to Orthogonal Polynomials
View Description Hide DescriptionQuantum systems consisting of a linear chain of n harmonic oscillators coupled by a quadratic nearest‐neighbour interaction are considered. We investigate when such a system is analytically solvable, in the sense that the eigenvalues and eigenvectors of the interaction matrix have analytically closed expressions. This leads to a relation with Jacobi matrices of systems of discrete orthogonal polynomials. Our study is first performed in the case of canonical quantization. Then we consider these systems under Wigner quantization, leading to solutions in terms of representations of Lie superalgebras. Finally, we show how such analytically solvable Hamiltonians also play a role in another application, that of spin chains used as communication channels in quantum computing. In this context, the analytic solvability leads to closed form expressions for certain transition amplitudes.

The Centrally Extended Heisenberg Algebra and Its Connection with the Schrödinger, Galilei and Renormalized Higher Powers of Quantum White Noise Lie Algebras
View Description Hide DescriptionIn previous papers we have shown that the one mode Heisenberg algebra Heis(1) admits a unique non‐trivial central extensions CeHeis(1) which can be realized as a sub‐Lie‐algebra of the Schrödinger algebra, in fact the Galilei Lie algebra. This gives a natural family of unitary representations of CeHeis(1) and allows an explicit determination of the associated group by exponentiation. In contrast with Heis(1), the group law for CeHeis(1) is given by nonlinear (quadratic) functions of the coordinates. The vacuum characteristic and moment generating functions of the classical random variables canonically associated to CeHeis(1) are computed. The second quantization of CeHeis(1) is also considered.

Random Variables and Positive Definite Kernels Associated with the Schrödinger Algebra
View Description Hide DescriptionWe show that the Feinsilver‐Kocik‐Schott (FKS) kernel for the Schrödinger algebra is not positive definite. We show how the FKS Schrödinger kernel can be reduced to a positive definite one through a restriction of the defining parameters of the exponential vectors. We define the Fock space associated with the reduced FKS Schrödinger kernel. We compute the characteristic functions of quantum random variables naturally associated with the FKS Schrödinger kernel and expressed in terms of the renormalized higher powers of white noise (or RHPWN) Lie algebra generators.

The Hamiltonian and Classification of osp(12) Representations
View Description Hide DescriptionThe quantization of the simple one‐dimensional Hamiltonian is of interest for its mathematical properties rather than for its physical relevance. In fact, the Berry‐Keating conjecture speculates that a proper quantization of could yield a relation with the Riemann hypothesis. Motivated by this, we study the so‐called Wigner quantization of which relates the problem to representations of the Lie superalgebra osp(12). In order to know how the relevant operators act in representation spaces of osp(12), we study all unitary, irreducible *‐representations of this Lie superalgebra. Such a classification has already been made by J.W.B. Hughes, but we reexamine this classification using elementary arguments.

A Vertex Algebra Attached to the Flag Manifold and Lie Algebra Cohomology
View Description Hide DescriptionWe compute the cohomology vertex algebra to the effect that it is isomorphic to We then find the Friedan‐Martinec‐Shenker‐Borisov bosonization of and verify that the latter algebra vanishes nonperturbatively.

Cohomologies of Configuration Spaces and Higher‐Dimensional Polylogarithms in Renormalization Group Problems
View Description Hide DescriptionThe deviation from commutativity of the renormalization and the action of all linear partial differential operators is the main source of the anomalies in quantum field theory, including the renormalization group action. This deviation is characterized by certain “renormalization cocycles” that are related to cohomologies of the so called (ordered) configuration spaces. Cohomological differential equations that determine the renormalization cocycles up to the renormalization freedom are obtained. The solution of these equations requires introducing transcendental extensions related to higher‐dimensional polylogarithms.

Invariant Subalgebras of Ghost Systems
View Description Hide DescriptionThis note is an expository account of recent results due to the author on the structure of invariant subalgebras of ghost systems under reductive group actions. Our main result is that these vertex algebras are strongly finitely generated. The algebra plays a fundamental role in their structure.

Conditions for Validity of the Gell‐Mann Formula in the Case of and/or su(n) Algebras
View Description Hide DescriptionThe Gell‐Mann (decontraction) formula is an expression designed as an inverse to the Inönü‐Wigner Lie algebra contraction. Its merits are notable simplicity and a great potential significance for the Lie algebra/group representation theory, while its drawback is a lack of general validity as an operator expression. The applicability of Gell‐Mann’s formula to various algebras and their representations was only partially treated. The validity conditions of the Gell‐Mann formula for the case of and su(n) algebras are clarified.

Ginsparg‐Wilson Formulation of 2D SQCD with Exact Lattice Supersymmetry
View Description Hide DescriptionWe discuss on a lattice formulation of 2D SQCD preserving one of its supercharges. In particular, the overlap Dirac operator, which satisfies the Ginsparg‐Wilson relation, is introduced to the matter sector of the theory. It exactly realizes chiral flavor symmetry on lattice, to make possible to define the lattice action for general number of the flavors of fundamental and anti‐fundamental matter multiplets. Furthermore, superpotential terms can be introduced with exact holomorphic or anti‐holomorphic structure on lattice. It is applicable to the lattice formulation of matter multiplets charged only under the central U(1) (the overall U(1)) of the gauge group and then to lattice models for gauged linear sigma models with exactly preserving one supercharge and their chiral flavor symmetry.

Parafermions, Ternary Algebras and Their Associated Superspace
View Description Hide DescriptionParafermions of order two are shown to be the fundamental tool to construct ternary superspaces related to cubic extensions of the Poincaré algebra.

Effective Supersymmetry Based on Superalgebra
View Description Hide DescriptionA minimal superalgebra S is proposed showing quarks, mesons, baryons and diquarks can all be viewed as its elements. This is a new minimal hadronic supersymmetry generalizing our earlier work based on SU (6/21). In the new scheme exotic mesons do not appear. Three body Hamiltonian for an excited baryon is shown to reduce to a two body quark‐diquark picture and is invariant under this algebra. Resulting two body Hamiltonian is exactly solved by means of confluent hypergeometric functions, giving rise to new mass formulae in remarkable agreement with experiments. Relationship to spectrum generating algebras is presented.