61 research outputs found
Instanton Expansion of Noncommutative Gauge Theory in Two Dimensions
We show that noncommutative gauge theory in two dimensions is an exactly
solvable model. A cohomological formulation of gauge theory defined on the
noncommutative torus is used to show that its quantum partition function can be
written as a sum over contributions from classical solutions. We derive an
explicit formula for the partition function of Yang-Mills theory defined on a
projective module for arbitrary noncommutativity parameter \theta which is
manifestly invariant under gauge Morita equivalence. The energy observables are
shown to be smooth functions of \theta. The construction of noncommutative
instanton contributions to the path integral is described in some detail. In
general, there are infinitely many gauge inequivalent contributions of fixed
topological charge, along with a finite number of quantum fluctuations about
each instanton. The associated moduli spaces are combinations of symmetric
products of an ordinary two-torus whose orbifold singularities are not resolved
by noncommutativity. In particular, the weak coupling limit of the gauge theory
is independent of \theta and computes the symplectic volume of the moduli space
of constant curvature connections on the noncommutative torus.Comment: 52 pages LaTeX, 1 eps figure, uses espf. V2: References added and
repaired; V3: Typos corrected, some clarifying explanations added; version to
be published in Communications in Mathematical Physic
Wilson Loops and Area-Preserving Diffeomorphisms in Twisted Noncommutative Gauge Theory
We use twist deformation techniques to analyse the behaviour under
area-preserving diffeomorphisms of quantum averages of Wilson loops in
Yang-Mills theory on the noncommutative plane. We find that while the classical
gauge theory is manifestly twist covariant, the holonomy operators break the
quantum implementation of the twisted symmetry in the usual formal definition
of the twisted quantum field theory. These results are deduced by analysing
general criteria which guarantee twist invariance of noncommutative quantum
field theories. From this a number of general results are also obtained, such
as the twisted symplectic invariance of noncommutative scalar quantum field
theories with polynomial interactions and the existence of a large class of
holonomy operators with both twisted gauge covariance and twisted symplectic
invariance.Comment: 23 page
Open Wilson Lines and Group Theory of Noncommutative Yang-Mills Theory in Two Dimensions
The correlation functions of open Wilson line operators in two-dimensional
Yang-Mills theory on the noncommutative torus are computed exactly. The
correlators are expressed in two equivalent forms. An instanton expansion
involves only topological numbers of Heisenberg modules and enables extraction
of the weak-coupling limit of the gauge theory. A dual algebraic expansion
involves only group theoretic quantities, winding numbers and translational
zero modes, and enables analysis of the strong-coupling limit of the gauge
theory and the high-momentum behaviour of open Wilson lines. The dual
expressions can be interpreted physically as exact sums over contributions from
virtual electric dipole quanta.Comment: 37 pages. References adde
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