Branislav Jurčo

Construction of non-Abelian gauge theories on noncommutative spaces

Abstract. We present a formalism to explicitly construct non-Abelian gauge theories on noncommutative spaces (induced via a star product with a constant Poisson tensor) from a consistency relation. This results in an expansion of the gauge parameter, the noncommutative gauge potential and fields in the fundamental representation, in powers of a parameter of the noncommutativity. This allows the explicit construction of actions for these gauge theories.

Enveloping algebra valued gauge transformations for non-abelian gauge groups on non-commutative spaces

Abstract. An enveloping algebra-valued gauge field is constructed, its components are functions of the Lie algebra-valued gauge field and can be constructed with the Seiberg-Witten map. This allows the formulation of a dynamics for a finite number of gauge field components on non-commutative spaces.

The standard model on non-commutative space-time

Abstract. We consider the standard model on a non-commutative space and expand the action in the non-commutativity parameter

Nonabelian noncommutative gauge theory via noncommutative extra dimensions

\theta^{\mu \nu}

Differential calculus on quantized simple lie groups

. No new particles are introduced; the structure group is

Noncommutative Yang-Mills from equivalence of star products

SU(3)\times SU(2)\times U(1)

Noncommutative gauge theory for Poisson manifolds

. We derive the leading order action. At zeroth order the action coincides with the ordinary standard model. At leading order in

Nonabelian Bundle Gerbes, Their Differential Geometry and Gauge Theory

\theta^{\mu\nu}

On coherent states for the simplest quantum groups

we find new vertices which are absent in the standard model on commutative space-time. The most striking features are couplings between quarks, gluons and electroweak bosons and many new vertices in the charged and neutral currents. We find that parity is violated in non-commutative QCD. The Higgs mechanism can be applied. QED is not deformed in the minimal version of the NCSM to the order considered.

Noncommutative Line Bundle and Morita Equivalence

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 Kontsevichs 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.