3,150 research outputs found
Construction of Field Algebras with Quantum Symmetry from Local Observables
It has been discussed earlier that ( weak quasi-) quantum groups allow for
conventional interpretation as internal symmetries in local quantum theory.
From general arguments and explicit examples their consistency with (braid-)
statistics and locality was established. This work addresses to the
reconstruction of quantum symmetries and algebras of field operators. For every
algebra \A of observables satisfying certain standard assumptions, an
appropriate quantum symmetry is found. Field operators are obtained which act
on a positive definite Hilbert space of states and transform covariantly under
the quantum symmetry. As a substitute for Bose/Fermi (anti-) commutation
relations, these fields are demonstrated to obey local braid relation.Comment: 50 pages, HUTMP 93-B33
Lectures on mathematical aspects of (twisted) supersymmetric gauge theories
Supersymmetric gauge theories have played a central role in applications of
quantum field theory to mathematics. Topologically twisted supersymmetric gauge
theories often admit a rigorous mathematical description: for example, the
Donaldson invariants of a 4-manifold can be interpreted as the correlation
functions of a topologically twisted N=2 gauge theory. The aim of these
lectures is to describe a mathematical formulation of partially-twisted
supersymmetric gauge theories (in perturbation theory). These partially twisted
theories are intermediate in complexity between the physical theory and the
topologically twisted theories. Moreover, we will sketch how the operators of
such a theory form a two complex dimensional analog of a vertex algebra.
Finally, we will consider a deformation of the N=1 theory and discuss its
relation to the Yangian, as explained in arXiv:1308.0370 and arXiv:1303.2632.Comment: Notes from a lecture series by the first author at the Les Houches
Winter School on Mathematical Physics in 2012. To appear in the proceedings
of this conference. Related to papers arXiv:1308.0370, arXiv:1303.2632, and
arXiv:1111.423
Seiberg-Witten maps and noncommutative Yang-Mills theories for arbitrary gauge groups
Seiberg-Witten maps and a recently proposed construction of noncommutative
Yang-Mills theories (with matter fields) for arbitrary gauge groups are
reformulated so that their existence to all orders is manifest. The ambiguities
of the construction which originate from the freedom in the Seiberg-Witten map
are discussed with regard to the question whether they can lead to inequivalent
models, i.e., models not related by field redefinitions.Comment: 12 pages; references added, minor misprints correcte
Gauge theories on the noncommutative sphere
Gauge theories are formulated on the noncommutative two-sphere. These
theories have only finite number of degrees of freedom, nevertheless they
exhibit both the gauge symmetry and the SU(2) "Poincar\'e" symmetry of the
sphere. In particular, the coupling of gauge fields to chiral fermions is
naturally achieved.Comment: 33 pages, LaTe
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