49 research outputs found
Quantum Algebras and the Homological Perturbation Lemma
Quantum algebras are a generalization of algebras with
a scalar product and with operations corresponding to higher genus graphs. We
construct a minimal model of a given quantum algebra via the
homological perturbation lemma and show that it's given by a Feynman diagram
expansion, computing the effective action in the finite-dimensional
Batalin-Vilkovisky formalism. We also construct a homotopy between the original
and this effective quantum algebra.Comment: 27 pages, fixed typos and the section 4.
Twisted Quantum Lax Equations
We give the construction of twisted quantum Lax equations associated with
quantum groups. We solve these equations using factorization properties of the
corresponding quantum groups. Our construction generalizes in many respects the
Adler-Kostant-Symes construction for Lie groups and the construction of M. A.
Semenov Tian-Shansky for the Lie-Poisson case.Comment: 23 pages, late
L∞-Algebras of Classical Field Theories and the Batalin-Vilkovisky Formalism
We review in detail the Batalin–Vilkovisky formalism for Lagrangian field theories and its mathematical foundations with an emphasis on higher algebraic structures and classical field theories. In particular, we show how a field theory gives rise to an L∞‐algebra and how quasi‐isomorphisms between L∞‐algebras correspond to classical equivalences of field theories. A few experts may be familiar with parts of our discussion, however, the material is presented from the perspective of a very general notion of a gauge theory. We also make a number of new observations and present some new results. Most importantly, we discuss in great detail higher (categorified) Chern–Simons theories and give some useful shortcuts in usually rather involved computations
Crossed Module Bundle Gerbes; Classification, String Group and Differential Geometry
We discuss nonabelian bundle gerbes and their differential geometry using
simplicial methods. Associated to any crossed module there is a simplicial
group NC, the nerve of the 1-category defined by the crossed module and its
geometric realization |NC|. Equivalence classes of principal bundles with
structure group |NC| are shown to be one-to-one with stable equivalence classes
of what we call crossed module gerbes bundle gerbes. We can also associate to a
crossed module a 2-category C'. Then there are two equivalent ways how to view
classifying spaces of NC-bundles and hence of |NC|-bundles and crossed module
bundle gerbes. We can either apply the W-construction to NC or take the nerve
of the 2-category C'. We discuss the string group and string structures from
this point of view. Also a simplicial principal bundle can be equipped with a
simplicial connection and a B-field. It is shown how in the case of a
simplicial principal NC-bundle these simplicial objects give the bundle gerbe
connection and the bundle gerbe B-field
Noncommutative gauge theory for Poisson manifolds
A noncommutative gauge theory is associated to every Abelian gauge theory on
a Poisson manifold. The semi-classical and full quantum version of the map from
the ordinary gauge theory to the noncommutative gauge theory (Seiberg-Witten
map) is given explicitly to all orders for any Poisson manifold in the Abelian
case. In the quantum case the construction is based on Kontsevich's formality
theorem.Comment: 12 page
More on quantum groups from the the quantization point of view
Star products on the classical double group of a simple Lie group and on
corresponding symplectic grupoids are given so that the quantum double and the
"quantized tangent bundle" are obtained in the deformation description.
"Complex" quantum groups and bicovariant quantum Lie algebras are discused from
this point of view. Further we discuss the quantization of the Poisson
structure on symmetric algebra leading to the quantized enveloping
algebra as an example of biquantization in the sense of Turaev.
Description of in terms of the generators of the bicovariant
differential calculus on is very convenient for this purpose. Finally
we interpret in the deformation framework some well known properties of compact
quantum groups as simple consequences of corresponding properties of classical
compact Lie groups. An analogue of the classical Kirillov's universal character
formula is given for the unitary irreducible representation in the compact
case.Comment: 18 page
Nonabelian noncommutative gauge theory via noncommutative extra dimensions
The concept of covariant coordinates on noncommutative spaces leads directly
to gauge theories with generalized noncommutative gauge fields of the type that
arises in string theory with background B-fields. The theory is naturally
expressed in terms of cochains in an appropriate cohomology; we discuss how it
fits into the framework of projective modules. The equivalence of star products
that arise from the background field with and without fluctuations and
Kontsevich's formality theorem allow an explicitly construction of a map that
relates ordinary gauge theory and noncommutative gauge theory (Seiberg-Witten
map.) As application we show the exact equality of the Dirac-Born-Infeld action
with B-field in the commutative setting and its semi-noncommutative cousin in
the intermediate picture. Using noncommutative extra dimensions the
construction is extended to noncommutative nonabelian gauge theory for
arbitrary gauge groups; an explicit map between abelian and nonabelian gauge
fields is given. All constructions are also valid for non-constant B-field,
Poisson structure and metric.Comment: 38 pages, 1 figure; v2: references adde