Geometry, if understood properly, is still the closest link between mathematics and theoretical physics, even for quantum concepts. In this collection of outstanding survey articles the concept of non-commutation geometry and the idea of quantum groups are discussed from various points of view. Furthermore the reader will find contributions to conformal field theory and to superalgebras and supermanifolds. The book addresses both physicists and mathematicians.
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Higgs fields and superconnections.- Noncommutative differential
geometry, quantum mechanics and gauge theory.- to non-commutative geometry
and Yang-Mills model-building.- II. Gauge-field model-building via
non-commutative differential geometry.- Measuring coalgebras, quantum
group-like objects, and non-commutative geometry.- Tensor Operator Structures
in Quantum Unitary Groups.- Quantum groups and quantum complete
integrability: Theory and experiment.- Some ideas and results on integrable
nonlinear evolution systems.- An algebraic characterization of complete
integrability for Hamiltonian systems.- Integrable lattice models and their
scaling limits QFT and CFT.- Quantum groups, Riemann surfaces and conformal
field theory.- Some physical applications of category theory.- From poisson
groupoids to quantum groupoids and back.- Quantization on Kähler manifolds.-
A new class of infinite-dimensional Lie algebras (continuum Lie algebras) and
associated nonlinear systems.- Exchange Algebra in the Conformal Affine sl 2
Toda Field Theory.- Some properties of p-lines.- Breaking of supersymmetry
through anomalies in composite spinor operators.- Conformal field theory and
moduli spaces of vector bundles over variable Riemann surfaces.- Instanton
homology.- W- geometry.- Connections between CFT and topology via Knot
theory.- Stochastic calculus in superspace and supersymmetric Hamiltonians.-
Geometric models and the modulli spaces for string theories.- Supersymmetric
products of SUSY-curves °.- Classical superspaces and related structures.-
Remarks on the differential identities in Schouten-Nijenhuis algebra.-
Generic irreducible representations of classical Lie superalgebras.-
Krichever construction of solutions to the super KP hierarchies.- The
structure of supersymplecticsupermanifolds.- Gauge fixing: Geometric and
probabilistic aspects of yang-mills gauge theories.- A renormalizable theory
of quantum gravity.- Third order nonlinear Hamiltonian systems: Some remarks
on the the action-angle transformation.- Tensor products of q p = 1 quantum
groups and WZW fusion rules.- The modular group and super-KMS functionals.-
New quantum representation for gravity and Yang-Mills theory.- Geometric
quantization of the five-dimensional Kepler problem.- Structure functions on
the usual and exotic symplectic and periplectic supermanifolds.- Symbols
alias generating functionals a supergeometric point of view.- Sheaves of
graded Lie algebras over variable Riemann surfaces and a paired
Weil-Petersson inner product.
