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Noncommutative Geometry (1994)

by A Connes
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String theory and noncommutative geometry

by Nathan Seiberg, Edward Witten - JHEP , 1999
"... We extend earlier ideas about the appearance of noncommutative geometry in string theory with a nonzero B-field. We identify a limit in which the entire string dynamics is described by a minimally coupled (supersymmetric) gauge theory on a noncommutative space, and discuss the corrections away from ..."
Abstract - Cited by 794 (8 self) - Add to MetaCart
We extend earlier ideas about the appearance of noncommutative geometry in string theory with a nonzero B-field. We identify a limit in which the entire string dynamics is described by a minimally coupled (supersymmetric) gauge theory on a noncommutative space, and discuss the corrections away from this limit. Our analysis leads us to an equivalence between ordinary gauge fields and noncommutative gauge fields, which is realized by a change of variables that can be described explicitly. This change of variables is checked by comparing the ordinary Dirac-Born-Infeld theory with its noncommutative counterpart. We obtain a new perspective on noncommutative gauge theory on a torus, its T-duality, and Morita equivalence. We also discuss the D0/D4 system, the relation to M-theory in DLCQ, and a possible noncommutative version of the six-dimensional (2, 0) theory. 8/99
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...t has been studied by many authors both from a mathematical and a physical perspective. The theory of operator algebras has been suggested as a framework for physics in noncommutative spacetime – see =-=[2]-=- for an exposition of the philosophy – and Yang-Mills theory on a noncommutative torus has been proposed as an example [3]. Though this example at first sight appears to be neither covariant nor causa...

Seiberg-Witten prepotential from instanton counting

by Nikita A. Nekrasov , 2002
"... In my lecture I consider integrals over moduli spaces of supersymmetric gauge field configurations (instantons, Higgs bundles, torsion free sheaves). The applications are twofold: physical and mathematical; they involve supersymmetric quantum mechanics of D-particles in various dimensions, direct co ..."
Abstract - Cited by 496 (9 self) - Add to MetaCart
In my lecture I consider integrals over moduli spaces of supersymmetric gauge field configurations (instantons, Higgs bundles, torsion free sheaves). The applications are twofold: physical and mathematical; they involve supersymmetric quantum mechanics of D-particles in various dimensions, direct computation of the celebrated Seiberg-Witten prepotential, sum rules for the solutions of the Bethe ansatz equations and their relation to the Laumon’s nilpotent cone. As a by-product we derive some combinatoric identities involving the sums over Young tableaux.

Introduction to Modern Canonical Quantum General Relativity

by T. Thiemann , 2001
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Abstract - Cited by 416 (26 self) - Add to MetaCart
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On conformal field theories

by Michael R. Douglas, Nikita A. Nekrasov - in fourdimensions,” Nucl. Phys. B533 , 1998
"... We review the generalization of field theory to space-time with noncommuting coordinates, starting with the basics and covering most of the active directions of research. Such theories are now known to emerge from limits of M theory and string theory, and to describe quantum Hall states. In the last ..."
Abstract - Cited by 365 (0 self) - Add to MetaCart
We review the generalization of field theory to space-time with noncommuting coordinates, starting with the basics and covering most of the active directions of research. Such theories are now known to emerge from limits of M theory and string theory, and to describe quantum Hall states. In the last few years they have been studied intensively, and many qualitatively new phenomena have been discovered, both on the classical and quantum level. Submitted to Reviews of Modern Physics.
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...tive geometry. Limitations on length would not permit more than the most cursory introduction to this subject here, and since so many introductions are already available, starting with the excellent (=-=Connes 1994-=-), much of which is quite readable by physicists, and including (Connes 1995, 2000b,a; Gracia-Bondia et al. 2001; Douglas 1999) as well as the reviews cited in the introduction, we will content oursel...

Noncommutative Perturbative Dynamics

by Shiraz Minwalla, Mark Van Raamsdonk - hep-th/9912072; M. Van Raamsdonk and N. Seiberg, “Comments On Noncommutative Perturbative Dynamics,” JHEP 0003 (2000) 035, hep-th/0002186
"... We study the perturbative dynamics of noncommutative field theories on Rd, and find an intriguing mixing of the UV and the IR. High energies of virtual particles in loops produce non-analyticity at low momentum. Consequently, the low energy effective action is singular at zero momentum even when the ..."
Abstract - Cited by 364 (1 self) - Add to MetaCart
We study the perturbative dynamics of noncommutative field theories on Rd, and find an intriguing mixing of the UV and the IR. High energies of virtual particles in loops produce non-analyticity at low momentum. Consequently, the low energy effective action is singular at zero momentum even when the original noncommutative field theory is massive. Some of the nonplanar diagrams of these theories are divergent, but we interpret these divergences as IR divergences and deal with them accordingly. We explain how this UV/IR mixing arises from the underlying noncommutativity. This phenomenon is reminiscent of the channel duality of the double twist diagram in open string theory.
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...mutative R d and in most of the note we discuss only scalar field theories (we will briefly mention gauge theories in the last section). Although noncommutative gauge theories appear in string theory =-=[13]-=- (see [14] and references therein for recent developments), through most of the paper our discussion will be field theoretic. The underlying R d is labeled by d noncommuting coordinates satisfying [x ...

Gravity coupled with matter and the foundation of non commutative geometry

by Alain Connes , 1996
"... We first exhibit in the commutative case the simple algebraic relations between the algebra of functions on a manifold and its infinitesimal length element ds. Its unitary representations correspond to Riemannian metrics and Spin structure while ds is the Dirac propagator ds = ×— × = D −1 where D i ..."
Abstract - Cited by 343 (17 self) - Add to MetaCart
We first exhibit in the commutative case the simple algebraic relations between the algebra of functions on a manifold and its infinitesimal length element ds. Its unitary representations correspond to Riemannian metrics and Spin structure while ds is the Dirac propagator ds = ×— × = D −1 where D is the Dirac operator. We extend these simple relations to the non commutative case using Tomita’s involution J. We then write a spectral action, the trace of a function of the length element in Planck units, which when applied to the non commutative geometry of the Standard Model will be shown (in a joint work with Ali Chamseddine) to give the SM Lagrangian coupled to gravity. The internal fluctuations of the non commutative geometry are trivial in the commutative case but yield the full bosonic sector of SM with all correct quantum numbers in the slightly non commutative case. The group of local gauge transformations appears spontaneously as a normal subgroup of the diffeomorphism group.

The quantum structure of spacetime at the Planck scale and quantum fields

by Sergio Doplicher, Klaus Fredenhagen, John E. Roberts - COMMUN. MATH. PHYS. 172, 187–220 (1995) , 1995
"... We propose uncertainty relations for the different coordinates of spacetime events, motivated by Heisenberg’s principle and by Einstein’s theory of classical gravity. A model of Quantum Spacetime is then discussed where the commutation relations exactly implement our uncertainty relations. We outl ..."
Abstract - Cited by 332 (6 self) - Add to MetaCart
We propose uncertainty relations for the different coordinates of spacetime events, motivated by Heisenberg’s principle and by Einstein’s theory of classical gravity. A model of Quantum Spacetime is then discussed where the commutation relations exactly implement our uncertainty relations. We outline the definition of free fields and interactions over QST and take the first steps to adapting the usual perturbation theory. The quantum nature of the underlying spacetime replaces a local interaction by a specific nonlocal effective interaction in the ordinary Minkowski space. A detailed study of interacting QFT and of the smoothing of ultraviolet divergences is deferred to a subsequent paper. In the classical limit where the Planck length goes to zero, our Quantum Spacetime reduces to the ordinary Minkowski space times a two component space whose components are homeomorphic to the tangent bundle TS 2 of the 2–sphere. The relations with Connes’ theory of the standard model will be studied elsewhere.
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...tion on ordinary Minkowski space where the nonlocal corrections are at least quadratic in λP. Gauge theories on the quantum spacetime should be formulated in the framework of non–commutative geometry =-=[8]-=-. More substantial deviations from theories on classical spacetime are to be expected; quantum electrodynamics, for example, will be a non–Abelian gauge theory. The occurrence of the two point set {1,...

An introduction to noncommutative spaces and their geometry

by Giovanni Landi , 1997
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Abstract - Cited by 259 (18 self) - Add to MetaCart
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An Introduction to Noncommutative Geometry

by Joseph C. Várilly
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Abstract - Cited by 240 (18 self) - Add to MetaCart
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...homology of algebras; thus the skeleton of the noncommutative torus will consist of a 0-cocycle, two 1-cocycles and a 2-cocycle on the algebra Aθ. The appropriate theory for that is cyclic cohomology =-=[9, 19, 22]-=-. It is a topological theory insofar as it depends only on the algebra Aθ and not on the geometries determined by its K-cycles. Definition. A cyclic n-cochain over an algebra A is an (n + 1)-linear fo...

Strings in background electric field, space/time noncommutativity and a new noncritical string theory

by N. Seiberg, L. Susskind, N. Toumbas , 2000
"... Searching for space/time noncommutativity we reconsider open strings in a constant background electric field. The main difference between this situation and its magnetic counterpart is that here there is a critical electric field beyond which the theory does not make sense. We show that this critica ..."
Abstract - Cited by 177 (2 self) - Add to MetaCart
Searching for space/time noncommutativity we reconsider open strings in a constant background electric field. The main difference between this situation and its magnetic counterpart is that here there is a critical electric field beyond which the theory does not make sense. We show that this critical field prevents us from finding a limit in which the theory becomes a field theory on a noncommutative spacetime. However, an appropriate limit toward the critical field leads to a novel noncritical string theory on a noncommutative spacetime.
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