String Geometry and the Noncommutative Torus |
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Authors: | Giovanni Landi Fedele Lizzi Richard J Szabo |
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Institution: | (1) Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK, UK;(2) Dipartimento di Scienze Fisiche, Università di Napoli Federico II and INFN, Sezione di Napoli, Mostra d'Oltremare Pad. 20, 80125 Napoli, Italy. E-mail: fedele.lizzi@na.infn.it, IT;(3) Department of Physics – Theoretical Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP, UK, UK |
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Abstract: | We construct a new gauge theory on a pair of d-dimensional noncommutative tori. The latter comes from an intimate relationship between the noncommutative geometry associated
with a lattice vertex operator algebra ? and the noncommutative torus. We show that the tachyon algebra of ? is naturally
isomorphic to a class of twisted modules representing quantum deformations of the algebra of functions on the torus. We construct
the corresponding real spectral triples and determine their Morita equivalence classes using string duality arguments. These
constructions yield simple proofs of the O(d,d;ℤ) Morita equivalences between d-dimensional noncommutative tori and give a natural physical interpretation of them in terms of the target space duality group
of toroidally compactified string theory. We classify the automorphisms of the twisted modules and construct the most general
gauge theory which is invariant under the automorphism group. We compute bosonic and fermionic actions associated with these
gauge theories and show that they are explicitly duality-symmetric. The duality-invariant gauge theory is manifestly covariant
but contains highly non-local interactions. We show that it also admits a new sort of particle-antiparticle duality which
enables the construction of instanton field configurations in any dimension. The duality non-symmetric on-shell projection
of the field theory is shown to coincide with the standard non-abelian Yang–Mills gauge theory minimally coupled to massive
Dirac fermion fields.
Received: 26 October 1998/ Accepted: 9 April 1999 |
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